Using the distributive property when multiplying

WEBVTT
Kind: captions
Language: en

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- [Instructor] What we're
going to do in this video

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is dig a little bit deeper

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into our understanding of multiplication.

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And just as an example,

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we're going to use four times seven.

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And some of you might know
what four times seven is,

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but even in this case,

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I think you might get
something from this video

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because we're gonna think
about how you can break down

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a multiplication question
into simpler parts,

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and that's going to be useful
well beyond four times seven.

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It's going to be useful in your future

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when you're tackling more
and more complicated things.

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Now there's a couple of
ways that we can visualize

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four times seven.

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My favorite way is to
visualize it with angry cats,

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so let's bring on the angry cats.

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(imitates cats meowing)

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Yep, they're still angry.

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And we can see that
this is a representation

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of four times seven.

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We have four rows right over here.

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Four rows,

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and each of those rows have seven cats.

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And so you can see that right over here,

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each of those rows has seven cats.

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Some people would call
this a four by seven grid

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or four by seven array,

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however you want to view it,

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but if someone were to ask
you what's the total number

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of cats, it would be four
rows times seven columns,

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four times seven.

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Now another way to
represent four times seven

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is also with a tape diagram.

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You might see something like this,

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where here, we're visualizing
it as seven fours,

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or you could view it as
four plus four plus four

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plus four plus four plus four plus four.

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Now that's all well and good
and you can add that up,

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if you like,

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but what I promised you is
that we would figure out ways

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to break down things

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that might simplify things in the future.

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Well what if you didn't know
what four times seven is,

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but you knew what four times five is

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and you knew what four times two is?

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Well what's interesting is
that seven is five plus two,

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so what if we tried to first
figure out this many cats,

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so four rows and five
columns right over there,

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and then we tried to figure out

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this many cats?

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Four rows and two columns.

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And you can see that it's the
exact same number of cats.

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So one way to think about
it is four times seven

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is the exact same thing as four times,

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and I'm gonna use parentheses,
and that just means

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to do that part first,
is equal to four times,

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instead of seven,

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I could write that as five plus two,

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'cause that's what seven is,

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so all this is saying is four time seven

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is the same thing as
four times five plus two,

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where you do the five plus two first,

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'cause we have those
parentheses around it,

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and five plus two is,
indeed, equal to seven,

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and we can see that that is
equivalent to the total number

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of cats that we have here,

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which we could view as
what we just circled off

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in this orangeish pink color

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which would be four rows of five,

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so that would be equal to four times five.

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Four times five,

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and then to that, we can add
this second group of cat heads

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or angry cat heads,

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and that is four rows of two.

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So that's four times two,

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and we could put parentheses if we want,

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just to make it a little
bit more readable.

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Now why did I do that?

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Well, some folks might
find four times five

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a little bit more straightforward.

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I could skip count by five.

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I can go five, 10, 15, 20.

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Also four times two might be a

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little bit more straightforward,

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and so it could be easier to say hey,

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this is just going to be four times five,

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which is 20, plus four times
two, which is equal to eight,

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and so that is just
going to be equal to 28.

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And you could have thought
about it the same way down here

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with what is sometimes
called a tape diagram.

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We could say, all right,

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if I have five fours,

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that's this mount right over here.

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That is the four times the five,

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and then I could add
that to the two fours,

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the four times the two right over here,

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and that's another way to
get to four times seven.

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So the big picture here is
even if you're not dealing

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with four times seven,
if you're not dealing

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with angry cats, and in most of our lives,

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we actually try to avoid angry cats,

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there might be a way to
break down the numbers

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that you're multiplying into ones

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that you might be more familiar with.

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I'll give you one more example.

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Let's say someone were to ask you, well,

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what is six times nine?

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Pause this video and see
if you can break this down

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in some useful way.

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Well, maybe you know what six times 10 is,

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and you also know what six times one is,

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so you could rerwrite nine

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as 10 minus one.

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Well then this would
mean that six times nine

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is the same thing as
six times 10 minus one.

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Based on exactly what we just did up here,

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that says that this whole thing

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is going to be the same
thing as six times 10.

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Six times 10

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minus six times one.

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One way to think about it is,

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I just distributed the six.

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That's the distributive
property right over there,

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and then six times 10 is equal to 60,

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and then six times one is equal to six.

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And it might be easier for me to say, hey,

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16 minus six in my head,

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that's equal to 54.

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So I know what some of you are thinking.

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Six times nine seems so clean,

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and now I've involved all
of this other symbolism,

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symbols, and I've written
down more numbers,

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but at the end of the day,

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I'm trying to give you skills

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for breaking down problems

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and including ways that you might want

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to do it in your head.

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If you're like, hey, I'm kind of foggy

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on what six times nine is,

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but six times 10, hey, I know that's 60,

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and six times one, of course, that's six,

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well what if I view this
as six times 10 minus one?

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And then I could tackle it and get 54.

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And once again,

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you might know six times nine.

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You might know four times seven,

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but in the future,

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it might be useful for
bigger and bigger numbers

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to think about how
could I break this down?


Using associate property to simplify multiplication

WEBVTT
Kind: captions
Language: en

00:00:00.340 --> 00:00:01.860
- [Instructor] In this video,
we're gonna think about

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how we can use our knowledge

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of multiplying single digit
numbers to multiply things

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that might involve two digits.

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So for example, let's start
with, what is five times 18?

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And you can pause the video
and see how you might try

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to approach this, and
then we'll do it together.

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All right, so if we're trying
to tackle five times 18,

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one strategy could be to
say, hey, can I re-express 18

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as the product of two numbers?

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And the one that jumps
out at me is that 18

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is the same thing as two times nine,

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and so I could rewrite five times 18.

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This is the same thing as five

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times, instead of 18, I
can write two times nine.

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Now, why does this help us?

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Well, instead of multiplying
the two times nine first

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to get the 18, and then
multiplying that by five,

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what we could do is we
could multiply the five

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times the two first.

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And you might be thinking,
wait, wait, wait,

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hold on a second.

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Before, you did the two times nine first,

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and now, you're telling me

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that you're going to change the order?

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That you're going to say, hey,

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let's multiply the five times
two first, is that okay?

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And the simple answer is, yes, it is okay.

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If you are multiplying
a string of numbers,

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you can do them in any
order that you choose,

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and so this is often known

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as the associative
property of multiplication.

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We can associate the
two with the nine first.

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We can multiply those first,

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or we can have an association
with the five and the two.

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We can multiply those two first.

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Now, why is that helpful?

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Well, what is five times two?

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Well, that's pretty straightforward.

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That's going to be equal to 10.

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So this is going to be equal to 10.

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We're doing that same color.

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10 times nine.

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Now, 10 times nine is a
lot more straightforward

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for most of us than five times 18.

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10 times nine is equal to 90.

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Let's do another example.

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Let's say we wanna figure
out what three times 21 is.

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Pause this video and see if
you can work through that.

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There's multiple ways to do it,

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but see if you can do it the way

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we just approached this first example.

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Well, as you could image,
we want to re-express 21

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as the product of smaller numbers.

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So we could rewrite 21 as
three times seven maybe?

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And so if we rewrite it
as three times seven,

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and now, we do the
three times three first.

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So I'm just gonna put parenthesis
there, which we can do

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because the associative
property of multiplication.

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Fancy word for something

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that is hopefully a little bit intuitive.

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Well then, this is going to be equal to,

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what's three times three?

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It is nine,

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and then times seven,
which you may already know

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is equal to 63.

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Let's do another example.

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This is kinda fun.

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Let's say we wanna figure
out what 14 times five is.

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Pause this video and see
if you can figure that out.

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Well, we could once
again try to break up 14

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into the product of smaller numbers.

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14 is two times seven,

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so we can rewrite this as two times seven

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or as seven times two,

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and I'm writing it as seven times two,

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because I want to associate
the two with the five

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to get the 10 times five,

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and then I can multiply
the two times five first.

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And so this is going to
give us seven times 10,

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seven times 10, which is
of course equal to 70.

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One more example.

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Let's say we want to
calculate 15 times three.

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How would you tackle that?

00:03:55.690 --> 00:04:00.510
Well, we can break up 15
into five times three.

00:04:00.510 --> 00:04:02.540
Five times three,

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and then we can multiply that,
of course, by this three,

00:04:05.980 --> 00:04:08.570
and then we can multiply
the threes together first,

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and then this amounts to five times nine.

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And five times nine,

00:04:14.770 --> 00:04:17.110
you might already be familiar with this.

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This is going to be equal to 45.

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Another way to get to
45, you can say, hey,

00:04:22.600 --> 00:04:24.970
five times 10 is 50.

00:04:24.970 --> 00:04:27.590
So five times 9 is going
to be five less than that,

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which is also 45.


Division in context examples

WEBVTT
Kind: captions
Language: en

00:00:00.180 --> 00:00:02.650
- [Instructor] We are asked,
which problem can we solve

00:00:02.650 --> 00:00:05.890
with 42 divided by seven?

00:00:05.890 --> 00:00:08.930
And they explain three
different scenarios here.

00:00:08.930 --> 00:00:10.010
We need to pick one of them,

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so pause this video and have a go at it

00:00:12.270 --> 00:00:14.020
before we work through it together.

00:00:15.230 --> 00:00:16.830
All right, now let's work
through this together.

00:00:16.830 --> 00:00:17.970
So let's see, choice A says,

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Steven packed 42 candy bars in his bag.

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He gave seven of them away to his friends.

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How many candy bars does Steven have left?

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Well, this isn't a situation of dividing.

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This is a situation
where he starts with 42,

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and he takes seven of them away.

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So this would be the
expression 42 minus seven,

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not the expression 42 divided by seven.

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Rule that one out.

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Julie has seven bags of jellybeans.

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There are 42 jellybeans in one bag.

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How many jellybeans does Julie have?

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Well, for each of those seven bags,

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she has 42 jellybeans.

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So this would be seven times
42, not 42 divided by seven.

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Rule that one out.

00:01:01.990 --> 00:01:05.750
Leslie ran for 42 minutes
total in one week.

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If she ran the same number
of minutes each day,

00:01:08.570 --> 00:01:11.890
how many minutes did
Leslie run in one day?

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So she ran 42 minutes
in total in the week,

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and she ran the same amount every day.

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There's seven days in a week.

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So if we wanna figure out
how much did she run per day,

00:01:21.560 --> 00:01:23.530
you just divide the 42 minutes total

00:01:23.530 --> 00:01:25.650
divided by seven days in a week

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to figure out this would
tell you right over here,

00:01:27.840 --> 00:01:30.750
how many minutes Leslie ran each day.

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And so that is the problem
that we would solve

00:01:32.780 --> 00:01:35.690
with 42 divided by seven.

00:01:35.690 --> 00:01:37.223
Let's do another example.

00:01:38.140 --> 00:01:41.780
Here we are told Kaleb
has 21 action figures.

00:01:41.780 --> 00:01:44.700
He puts three action figures in each box.

00:01:44.700 --> 00:01:48.430
What does the expression 21
divided by three represent?

00:01:48.430 --> 00:01:49.760
So pause this video again,

00:01:49.760 --> 00:01:52.010
and see if you can answer that.

00:01:52.010 --> 00:01:53.300
All right, so this is a situation

00:01:53.300 --> 00:01:55.380
where we're starting with 21,

00:01:55.380 --> 00:01:59.640
and we are dividing it
into groups of three.

00:01:59.640 --> 00:02:04.640
So these are groups, groups of three.

00:02:05.610 --> 00:02:08.860
So this is going to tell
us how many groups we need,

00:02:08.860 --> 00:02:11.860
if we're going to divide
21 into groups of three.

00:02:11.860 --> 00:02:15.830
So if I wrote 21 divided by three,

00:02:15.830 --> 00:02:20.310
and so this is groups of
three, groups of three.

00:02:20.310 --> 00:02:21.890
This is going to be equal to something,

00:02:21.890 --> 00:02:24.020
and many of you might know what this is,

00:02:24.020 --> 00:02:26.840
but this is going to
tell us how many groups,

00:02:26.840 --> 00:02:28.070
how many groups.

00:02:28.070 --> 00:02:30.500
And here a box represents a group.

00:02:30.500 --> 00:02:32.800
How many groups?

00:02:32.800 --> 00:02:36.060
So this is going to be
the number of boxes.

00:02:36.060 --> 00:02:37.900
The number of action figures in each box,

00:02:37.900 --> 00:02:38.733
they already told us.

00:02:38.733 --> 00:02:39.680
There's three in each box,

00:02:39.680 --> 00:02:41.750
so that would be
represented just by a three.

00:02:41.750 --> 00:02:43.050
The total number of action figures,

00:02:43.050 --> 00:02:44.810
that would be 21.

00:02:44.810 --> 00:02:47.250
The number of boxes is the total number

00:02:47.250 --> 00:02:50.490
divided by how many in each box.

00:02:50.490 --> 00:02:51.540
Let's do another one.

00:02:53.300 --> 00:02:58.130
Alex, Brian, and Marta split
a box of 12 cookies evenly.

00:02:58.130 --> 00:03:00.250
Which expression helps us find out

00:03:00.250 --> 00:03:03.790
how many cookies each friend will receive?

00:03:03.790 --> 00:03:06.630
So pause this video again
and see if you can tackle it.

00:03:06.630 --> 00:03:09.000
All right, so we're splitting 12 cookies.

00:03:09.000 --> 00:03:10.190
So we'll start with 12,

00:03:10.190 --> 00:03:11.530
and we need to divide it,

00:03:11.530 --> 00:03:14.460
and how many people are
we dividing it between?

00:03:14.460 --> 00:03:16.870
Well we're dividing it between one, Alex,

00:03:16.870 --> 00:03:20.120
two, Brian, and three people, Marta.

00:03:20.120 --> 00:03:23.530
So we're going to divide
it into three equal groups,

00:03:23.530 --> 00:03:24.930
and you might know what that is,

00:03:24.930 --> 00:03:26.350
but we don't even have to figure that out.

00:03:26.350 --> 00:03:27.600
We just have to say, hey,

00:03:28.610 --> 00:03:31.740
how many cookies does each
of the three people receive?

00:03:31.740 --> 00:03:34.480
Well, it's going to be 12
divided by three cookies.

00:03:34.480 --> 00:03:36.900
So it is this choice right over here.

00:03:36.900 --> 00:03:39.150
Three divided by 12 doesn't
make sense in this situation.

00:03:39.150 --> 00:03:40.150
You don't have three cookies,

00:03:40.150 --> 00:03:42.060
and you're not dividing
it amongst 12 people,

00:03:42.060 --> 00:03:43.470
and C doesn't make sense.

00:03:43.470 --> 00:03:45.140
You're not just taking 12 cookies

00:03:45.140 --> 00:03:46.800
and dividing it between two people.

00:03:46.800 --> 00:03:48.833
You're dividing it between three people.


Visualizing division with arrays

WEBVTT
Kind: captions
Language: en

00:00:00.230 --> 00:00:02.230
- [Instructor] We have three
different pictures here,

00:00:02.230 --> 00:00:04.990
and my question to get us warmed up

00:00:04.990 --> 00:00:09.990
is which of these could
represent 20 divided by four?

00:00:10.730 --> 00:00:14.100
Pause this video and see
if you can figure that out.

00:00:14.100 --> 00:00:16.800
All right, so let's go
through each of these.

00:00:16.800 --> 00:00:18.120
And, actually, let's think about

00:00:18.120 --> 00:00:20.830
what each of them could represent.

00:00:20.830 --> 00:00:24.030
So how many total circles
do we have in this picture

00:00:24.030 --> 00:00:24.970
right over here?

00:00:24.970 --> 00:00:28.620
Let's see, we have one, two, three, four,

00:00:28.620 --> 00:00:30.840
and then we have four groups of four.

00:00:30.840 --> 00:00:34.620
So four, eight, 12, 16.

00:00:34.620 --> 00:00:37.230
So this looks like 16 circles.

00:00:37.230 --> 00:00:41.070
16 circles divided into,

00:00:41.070 --> 00:00:44.930
we could do it as divided
into groups of four,

00:00:44.930 --> 00:00:46.470
or we could actually also view it

00:00:46.470 --> 00:00:48.780
as divided into four equal groups,

00:00:48.780 --> 00:00:50.440
because both of those are true.

00:00:50.440 --> 00:00:53.620
And we see that that is
going to be equal to four.

00:00:53.620 --> 00:00:54.690
So this one over here,

00:00:54.690 --> 00:00:57.510
it's really representing
16 divided by four,

00:00:57.510 --> 00:00:59.790
not 20 divided by four.

00:00:59.790 --> 00:01:01.360
Now, what about this one?

00:01:01.360 --> 00:01:05.060
Let's see, here, it looks
like we have one, two,

00:01:05.060 --> 00:01:06.970
three, four, five groups

00:01:06.970 --> 00:01:10.140
of one, two, three, four, five.

00:01:10.140 --> 00:01:14.180
So this looks like 25 circles

00:01:14.180 --> 00:01:17.700
divided into groups of five,

00:01:17.700 --> 00:01:19.960
or divided into five equals groups,

00:01:19.960 --> 00:01:22.270
which is, of course, equal to five.

00:01:22.270 --> 00:01:26.193
So this is 25 divided by
five, not 20 divided by four.

00:01:27.040 --> 00:01:28.910
Now, but what about this one?

00:01:28.910 --> 00:01:32.330
So let's see, we have one,
two, three, four, five,

00:01:32.330 --> 00:01:37.330
and we have, one, two,
three, four groups of five.

00:01:38.520 --> 00:01:40.970
So five, 10, 15, 20.

00:01:40.970 --> 00:01:43.400
So we definitely have 20 circles there.

00:01:43.400 --> 00:01:46.810
And we have divided it
into four equal groups.

00:01:46.810 --> 00:01:49.180
We have divided it into four equal groups,

00:01:49.180 --> 00:01:51.220
and so that's exactly
what we're talking about,

00:01:51.220 --> 00:01:52.510
or what we started talking about.

00:01:52.510 --> 00:01:54.080
This is 20 divided by four.

00:01:54.080 --> 00:01:54.963
And you can see, when you take 20

00:01:54.963 --> 00:01:58.280
and divide it into four equal groups,

00:01:58.280 --> 00:02:03.280
then you get one, two, three,
four, five circles per group.

00:02:04.120 --> 00:02:05.013
And we're done.


What order to do operations in

WEBVTT
Kind: captions
Language: en

00:00:00.440 --> 00:00:02.210
- [Instructor] If I were to ask you,

00:00:02.210 --> 00:00:05.160
what is five

00:00:05.160 --> 00:00:09.850
minus three plus two?

00:00:09.850 --> 00:00:11.670
What would you say that is?

00:00:11.670 --> 00:00:14.503
Pause this video and
try to figure that out.

00:00:15.900 --> 00:00:17.640
All right, well if you
wanted to tackle this,

00:00:17.640 --> 00:00:19.810
you would really just read
it from left to right,

00:00:19.810 --> 00:00:21.780
or you would compute
it from left to right.

00:00:21.780 --> 00:00:25.010
You'd first figure out
what five minus three is,

00:00:25.010 --> 00:00:27.890
and five minus three is equal to two.

00:00:27.890 --> 00:00:30.450
And so you'd say, okay,
that's going to be two

00:00:30.450 --> 00:00:35.450
plus two, which of course
we know is equal to four.

00:00:36.160 --> 00:00:38.220
Now, what if I were to
involve a little bit

00:00:38.220 --> 00:00:40.300
of multiplication or division?

00:00:40.300 --> 00:00:43.380
If I were to go up to
you and I were to say,

00:00:43.380 --> 00:00:46.360
what is five

00:00:46.360 --> 00:00:50.660
plus three times two?

00:00:50.660 --> 00:00:52.860
Pause this video and
try to figure that out.

00:00:53.930 --> 00:00:57.350
Well, you might be tempted
to do it in a similar way.

00:00:57.350 --> 00:00:58.470
You might be tempted to say,

00:00:58.470 --> 00:01:00.400
okay, I'll just go from left to right.

00:01:00.400 --> 00:01:03.860
I'll first compute what five plus three is

00:01:03.860 --> 00:01:06.560
and that of course,
would be equal to eight,

00:01:06.560 --> 00:01:09.950
and then I would multiply eight times two

00:01:09.950 --> 00:01:12.100
to get 16.

00:01:12.100 --> 00:01:13.280
But it turns out that this

00:01:13.280 --> 00:01:15.960
would not be the right
way to approach this,

00:01:15.960 --> 00:01:17.740
because for very good reasons

00:01:17.740 --> 00:01:20.490
that we'll study in perhaps future videos,

00:01:20.490 --> 00:01:23.130
we don't always go left to right,

00:01:23.130 --> 00:01:24.230
especially when we're dealing

00:01:24.230 --> 00:01:26.490
with different types of operations.

00:01:26.490 --> 00:01:29.440
And the standard way
that this is approached

00:01:29.440 --> 00:01:32.390
is that multiplication
and division will be done

00:01:32.390 --> 00:01:34.860
before addition and subtraction.

00:01:34.860 --> 00:01:37.780
So the way that you would
actually compute this

00:01:37.780 --> 00:01:41.970
is that you would compute
the three times two first.

00:01:41.970 --> 00:01:43.600
So you would say that
this is the same thing

00:01:43.600 --> 00:01:46.160
as five plus six,

00:01:46.160 --> 00:01:51.160
five plus six, which is going
to be equal to 11, not 16.

00:01:54.070 --> 00:01:57.010
So this is going to be equal to 11.

00:01:57.010 --> 00:01:58.410
So with that out of the way,

00:01:58.410 --> 00:02:00.210
let's do a few more examples.

00:02:00.210 --> 00:02:03.810
Let's say someone were
to approach you and say,

00:02:03.810 --> 00:02:06.100
what is nine

00:02:06.100 --> 00:02:11.100
minus six divided by three?

00:02:11.130 --> 00:02:14.193
Pause this video and see
if you can figure that out.

00:02:15.400 --> 00:02:19.560
Well, as I just said, we would
do multiplication or division

00:02:19.560 --> 00:02:21.610
before we do subtraction.

00:02:21.610 --> 00:02:25.030
So we'll actually do the
six divided by three first,

00:02:25.030 --> 00:02:27.640
and six divided by three is two.

00:02:27.640 --> 00:02:31.830
So this simplifies to nine minus two,

00:02:31.830 --> 00:02:33.490
which is equal to seven,

00:02:33.490 --> 00:02:35.030
and I really wanna emphasize,

00:02:35.030 --> 00:02:37.320
it's important that you get this idea

00:02:37.320 --> 00:02:39.840
that multiplication and
division should be done

00:02:39.840 --> 00:02:42.240
before the addition or subtraction,

00:02:42.240 --> 00:02:45.120
unless you're given some
other types of directions.

00:02:45.120 --> 00:02:46.800
Because if you just
went left to right here,

00:02:46.800 --> 00:02:48.700
you would have said
nine minus six is three,

00:02:48.700 --> 00:02:50.300
and then you would've
divided that by three,

00:02:50.300 --> 00:02:52.400
which would have given you one,

00:02:52.400 --> 00:02:56.170
which is clearly a
different number than seven.

00:02:56.170 --> 00:02:59.770
Let's do one more example, just for kicks.

00:02:59.770 --> 00:03:03.020
Let's say that someone were to ask you,

00:03:03.020 --> 00:03:05.510
what is two

00:03:05.510 --> 00:03:09.550
times three minus one?

00:03:09.550 --> 00:03:12.580
Pause the video and try
to figure that one out.

00:03:12.580 --> 00:03:15.090
Well, in this one, doing
our multiplication first

00:03:15.090 --> 00:03:17.200
is also the same as going
left to right first.

00:03:17.200 --> 00:03:19.710
So if we do two times three first,

00:03:19.710 --> 00:03:22.200
two times three is six,

00:03:22.200 --> 00:03:23.260
so that's six,

00:03:23.260 --> 00:03:25.610
and then we would wanna
calculate six minus one,

00:03:25.610 --> 00:03:28.140
which is of course equal to five.

00:03:28.140 --> 00:03:31.730
So big picture, if you
just have a bunch of things

00:03:31.730 --> 00:03:34.590
that are being added, subtracted,
multiply, and divided,

00:03:34.590 --> 00:03:36.880
and there is no parenthesis in it,

00:03:36.880 --> 00:03:40.510
you would always prioritize
the multiplication

00:03:40.510 --> 00:03:44.303
or the division over the
addition or the subtraction.


Remainder theorem examples

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:02.130
- [Instructor] So we have the graph here

00:00:02.130 --> 00:00:04.570
of Y is equal to P of x,

00:00:04.570 --> 00:00:08.140
I could write it like
this, Y is equal to P of x.

00:00:08.140 --> 00:00:09.970
And they say, what is the remainder

00:00:09.970 --> 00:00:14.120
when P of x is divided by x plus three?

00:00:14.120 --> 00:00:17.370
So pause this video and see
if you can have a go at this

00:00:17.370 --> 00:00:20.180
and they tell us your
answer should be an integer.

00:00:20.180 --> 00:00:22.090
So, as you might have assumed,

00:00:22.090 --> 00:00:24.500
this will involve the
polynomial remainder theorem

00:00:24.500 --> 00:00:28.960
and all that tells us is that
hey if we were to take P of x

00:00:28.960 --> 00:00:31.723
and divide it by x plus three,

00:00:33.070 --> 00:00:35.000
whatever the remainder is here,

00:00:35.000 --> 00:00:37.600
so we'll say the remainder is equal to k.

00:00:37.600 --> 00:00:40.740
That value k is what we would have gotten

00:00:40.740 --> 00:00:42.620
if we took our polynomial

00:00:42.620 --> 00:00:46.210
and we evaluated it at the value of x

00:00:46.210 --> 00:00:48.760
that would have made x
plus three equal zero

00:00:48.760 --> 00:00:50.760
or just what would have happened

00:00:50.760 --> 00:00:54.400
if I evaluated our polynomial
as x equals negative three.

00:00:54.400 --> 00:00:55.610
You have to be very careful there

00:00:55.610 --> 00:00:56.950
sometimes people get confused.

00:00:56.950 --> 00:00:58.240
They see a positive three

00:00:58.240 --> 00:01:01.010
and then they evaluate the
polynomial at the positive three

00:01:01.010 --> 00:01:01.850
to figure out the remainder.

00:01:01.850 --> 00:01:02.683
No!

00:01:02.683 --> 00:01:03.910
If you saw a positive three there,

00:01:03.910 --> 00:01:06.350
you would evaluate the
polynomial at negative three

00:01:06.350 --> 00:01:10.700
but this should be equal to k as well.

00:01:10.700 --> 00:01:12.930
And so what is the remainder when P of x

00:01:12.930 --> 00:01:15.699
is divided by x plus three?

00:01:15.699 --> 00:01:20.260
Well it's going to be equal
to P of negative three.

00:01:20.260 --> 00:01:22.920
P of negative three it looks like

00:01:22.920 --> 00:01:24.760
it is equal to negative two.

00:01:24.760 --> 00:01:27.070
It is equal to negative
two, so our remainder

00:01:27.070 --> 00:01:30.510
is equal to negative
two in this situation.

00:01:30.510 --> 00:01:32.350
Let's do another example,

00:01:32.350 --> 00:01:35.040
actually let's do several more examples.

00:01:35.040 --> 00:01:39.060
Here we're told that P of x is
equal to all of this business

00:01:39.060 --> 00:01:42.590
where k is an unknown
integer, very interesting.

00:01:42.590 --> 00:01:47.270
P of x divided by x minus
two has a remainder of one.

00:01:47.270 --> 00:01:49.290
What is the value of k?

00:01:49.290 --> 00:01:52.700
So pause this video again,
see if you can work it out.

00:01:52.700 --> 00:01:55.230
All right, well this second sentence

00:01:55.230 --> 00:01:58.930
that P of x divided by x minus
two has a remainder of one,

00:01:58.930 --> 00:02:02.690
that tells us that P not of negative two

00:02:02.690 --> 00:02:04.360
but P of positive two,

00:02:04.360 --> 00:02:07.530
whatever x value would make
this expression equal zero.

00:02:07.530 --> 00:02:10.090
That P of two is equal to one.

00:02:10.090 --> 00:02:12.460
And then we could use this top information

00:02:12.460 --> 00:02:14.460
to figure out what P of two would be.

00:02:14.460 --> 00:02:17.400
It would be two to the fourth power

00:02:17.400 --> 00:02:22.130
minus two times two to the third power

00:02:22.130 --> 00:02:25.590
plus k times two squared.

00:02:25.590 --> 00:02:29.920
So, times two squared minus 11,

00:02:29.920 --> 00:02:32.400
and so all of that, that's
P of two right over here

00:02:32.400 --> 00:02:34.070
that's going to be equal to one.

00:02:34.070 --> 00:02:36.200
Two to the fourth is 16

00:02:36.200 --> 00:02:37.750
and then two times two to the third

00:02:37.750 --> 00:02:40.490
that's two to the fourth
again, so it's minus 16

00:02:40.490 --> 00:02:45.490
plus four k minus 11 is equal
to one, these cancel out.

00:02:45.960 --> 00:02:49.710
Now let's see we can add 11
to both sides of this equation

00:02:49.710 --> 00:02:54.710
and we get four k is equal to 12.

00:02:55.170 --> 00:02:59.950
Divide both sides by four and
we get k is equal to three

00:02:59.950 --> 00:03:01.010
and we're done.

00:03:01.010 --> 00:03:03.130
Let's do another example,
in fact let's do two more

00:03:03.130 --> 00:03:04.780
because we're having so much fun.

00:03:06.460 --> 00:03:11.230
So this next question tells
us, P of x is a polynomial

00:03:11.230 --> 00:03:15.420
and they tell us what P of x
divide by various things are,

00:03:15.420 --> 00:03:17.760
what the remainder would
be when you divide P of x

00:03:17.760 --> 00:03:19.540
by these various expressions.

00:03:19.540 --> 00:03:21.810
Find the following values of P of x,

00:03:21.810 --> 00:03:24.300
P of negative four and P of one.

00:03:24.300 --> 00:03:26.900
Pause this video and see
if you can have a go at it.

00:03:27.900 --> 00:03:31.960
All right, so P of negative four,

00:03:31.960 --> 00:03:34.610
this is going to be equal to the remainder

00:03:37.420 --> 00:03:42.380
when P of x divided by what.

00:03:43.460 --> 00:03:46.200
You might be tempted to say x minus four

00:03:46.200 --> 00:03:48.660
but they're trying to
trick you intentionally.

00:03:48.660 --> 00:03:50.350
This would be the remainder when P of x

00:03:50.350 --> 00:03:53.480
is divided by x plus four.

00:03:53.480 --> 00:03:54.990
And so they tell us right over here

00:03:54.990 --> 00:03:58.640
P of x divided by x plus four
has a remainder of three.

00:03:58.640 --> 00:04:00.270
So, it's going to be
three right over there.

00:04:00.270 --> 00:04:05.260
And similarly P of one, this
is going to be the remainder,

00:04:05.260 --> 00:04:10.260
this is the remainder
when P of x divided by

00:04:11.020 --> 00:04:14.650
not x plus one, but x minus one.

00:04:14.650 --> 00:04:16.880
So when P of x is divided by x minus one,

00:04:16.880 --> 00:04:19.430
the remainder is zero.

00:04:19.430 --> 00:04:21.743
Let's do one last example.

00:04:23.690 --> 00:04:26.480
So once again P of x is a polynomial

00:04:26.480 --> 00:04:28.940
and then they give us
a few values of P of x.

00:04:28.940 --> 00:04:30.550
And they say what is the remainder

00:04:30.550 --> 00:04:34.760
when P of x is divided by x minus three?

00:04:34.760 --> 00:04:37.120
Pause the video and try
to think about that.

00:04:37.120 --> 00:04:39.040
Well, we've gone over this multiple times,

00:04:39.040 --> 00:04:41.150
the remainder when P of x
is divided by x minus three

00:04:41.150 --> 00:04:45.580
that would be P of not negative
three, P of positive three.

00:04:45.580 --> 00:04:50.210
Whatever value of x makes this
entire expression equal zero.

00:04:50.210 --> 00:04:54.830
So P of positive three is equal to five.

00:04:54.830 --> 00:04:56.730
And similarly what is the remainder,

00:04:56.730 --> 00:04:58.450
actually not so similarly,
this is interesting.

00:04:58.450 --> 00:05:01.107
What is the remainder when
P of x is divided by x?

00:05:01.107 --> 00:05:02.030
I know what you're thinking

00:05:02.030 --> 00:05:05.460
it's like wait what
number am I dealing with?

00:05:05.460 --> 00:05:06.620
But if I were to rewrite this

00:05:06.620 --> 00:05:07.840
instead of saying divided by x,

00:05:07.840 --> 00:05:11.550
if I were to say divided by x plus zero

00:05:11.550 --> 00:05:13.480
then you'd be like oh now I get it.

00:05:13.480 --> 00:05:17.809
Or if I wrote divided by x minus zero,

00:05:17.809 --> 00:05:19.524
you'd be like oh, now I get it.

00:05:19.524 --> 00:05:23.150
This is going to be P
of and it doesn't matter

00:05:23.150 --> 00:05:25.240
whether I take a positive
or a negative zero,

00:05:25.240 --> 00:05:26.960
it's going to be P of zero.

00:05:26.960 --> 00:05:31.073
And P of zero, they tell us,
is negative one and we're done.


Dividing polynomials by x (no remainders)

WEBVTT
Kind: captions
Language: en

00:00:00.190 --> 00:00:01.770
- [Instructor] Let's say
someone walks up to you

00:00:01.770 --> 00:00:03.710
on the street, and they
give you this expression.

00:00:03.710 --> 00:00:08.710
X squared plus 7x plus
10 divided by x plus two.

00:00:10.300 --> 00:00:13.550
And they say, see if you
can simplify this thing.

00:00:13.550 --> 00:00:16.610
And so, pause this video
and see if you can do that.

00:00:16.610 --> 00:00:17.750
And one way to think about it is,

00:00:17.750 --> 00:00:22.450
what is x squared plus 7x
plus 10 divided by x plus two,

00:00:22.450 --> 00:00:23.700
what is that going to be?

00:00:24.860 --> 00:00:26.240
All right now, there's two ways

00:00:26.240 --> 00:00:27.330
that you could approach this.

00:00:27.330 --> 00:00:29.800
One way is to try to factor the numerator

00:00:29.800 --> 00:00:31.770
and see if it has a factor

00:00:31.770 --> 00:00:34.390
that is common to the denominator.

00:00:34.390 --> 00:00:35.700
So let's try to do that.

00:00:35.700 --> 00:00:37.540
So we've done this many, many times.

00:00:37.540 --> 00:00:39.110
If this looks new to you,

00:00:39.110 --> 00:00:41.100
I encourage you to review
factoring polynomials

00:00:41.100 --> 00:00:42.950
other places on Kahn Academy.

00:00:42.950 --> 00:00:44.900
But what two numbers add up to seven,

00:00:44.900 --> 00:00:47.180
and when you multiply em, you get 10?

00:00:47.180 --> 00:00:48.790
Well, that would be two and five.

00:00:48.790 --> 00:00:50.340
So we could rewrite that numerator

00:00:50.340 --> 00:00:54.600
as x plus two times x plus five.

00:00:54.600 --> 00:00:55.433
And then of course,

00:00:55.433 --> 00:00:59.090
the denominator, you
still have x plus two.

00:00:59.090 --> 00:01:01.800
And then we clearly see
we have a common factor.

00:01:01.800 --> 00:01:05.650
And so as long as x does
not equal negative two,

00:01:05.650 --> 00:01:07.310
because if x equals negative two,

00:01:07.310 --> 00:01:09.170
this whole expression is undefined,

00:01:09.170 --> 00:01:11.140
because then you get a
zero in the denominator.

00:01:11.140 --> 00:01:15.430
So as long as x does
not equal negative two,

00:01:15.430 --> 00:01:17.390
well then, we can divide the numerator

00:01:17.390 --> 00:01:19.840
and the denominator by x plus two.

00:01:19.840 --> 00:01:21.053
Once again, the reason why
I put that constraint is

00:01:21.053 --> 00:01:22.930
that we can't divide the
numerator and the denominator

00:01:22.930 --> 00:01:23.920
by zero.

00:01:23.920 --> 00:01:25.920
So for any other values of x,

00:01:25.920 --> 00:01:28.420
this x plus two will be non zero,

00:01:28.420 --> 00:01:29.510
and we could divide the numerator

00:01:29.510 --> 00:01:32.180
and the denominator by that,
and they would cancel out,

00:01:32.180 --> 00:01:36.540
and we would just be
left with x plus five.

00:01:36.540 --> 00:01:39.000
So another way to think
about it is this expression,

00:01:39.000 --> 00:01:41.170
our original expression,
could be viewed as

00:01:41.170 --> 00:01:46.170
x plus five for any x that
is not equal to negative two.

00:01:47.430 --> 00:01:50.060
Now the other way that
we could approach this

00:01:50.060 --> 00:01:53.310
is through algebraic long division,

00:01:53.310 --> 00:01:55.640
which is very analogous to
the type of long division

00:01:55.640 --> 00:01:56.880
that you might remember from,

00:01:56.880 --> 00:01:59.060
I believe it was, fourth grade.

00:01:59.060 --> 00:02:00.380
So what you do is you say, all right,

00:02:00.380 --> 00:02:02.390
I'm gonna divide x plus two

00:02:02.390 --> 00:02:07.390
into x squared plus seven x plus 10.

00:02:07.830 --> 00:02:10.790
And in this technique, you look
at the highest degree terms.

00:02:10.790 --> 00:02:13.340
So then, you have an x there
and an x squared there.

00:02:13.340 --> 00:02:16.310
And say, how many times
does x go into x squared?

00:02:16.310 --> 00:02:18.120
Well it goes x times.

00:02:18.120 --> 00:02:20.070
Now you'd write that in this column,

00:02:20.070 --> 00:02:22.830
because x is just x to the first power.

00:02:22.830 --> 00:02:24.950
You could view this as
the first degree column.

00:02:24.950 --> 00:02:27.890
It's analogous to the place values

00:02:27.890 --> 00:02:30.570
that we talk about when
we first learn numbers,

00:02:30.570 --> 00:02:33.790
or how we group or about place value.

00:02:33.790 --> 00:02:35.720
But here you could view
it as degree places

00:02:35.720 --> 00:02:36.740
or something like that.

00:02:36.740 --> 00:02:37.960
And then you take that x

00:02:37.960 --> 00:02:40.360
and you multiply it times
this entire expression.

00:02:40.360 --> 00:02:43.020
So x times two is 2x.

00:02:43.020 --> 00:02:44.860
Put that in the first degree column.

00:02:44.860 --> 00:02:48.440
X times x is x squared.

00:02:48.440 --> 00:02:49.650
And then what we wanna do

00:02:49.650 --> 00:02:51.930
is we wanna subtract
these things in yellow

00:02:51.930 --> 00:02:54.310
from what we originally had in blue.

00:02:54.310 --> 00:02:55.960
So we could do it this way.

00:02:55.960 --> 00:03:00.620
And then we will be left
with 7x minus 2x is 5x.

00:03:00.620 --> 00:03:03.930
And then x squared minus
x squared is just a zero.

00:03:03.930 --> 00:03:08.330
And then we can bring down this plus 10.

00:03:08.330 --> 00:03:11.080
And once again, we look at
the highest degree term.

00:03:11.080 --> 00:03:13.730
X goes into 5x five times.

00:03:13.730 --> 00:03:14.660
That's a zero degree.

00:03:14.660 --> 00:03:17.750
It's a constant, so I'll write
it in the constant column.

00:03:17.750 --> 00:03:19.990
Five times two is 10.

00:03:19.990 --> 00:03:22.430
Five times x is five.

00:03:22.430 --> 00:03:25.520
And then I'll subtract these
from what we have up here.

00:03:25.520 --> 00:03:27.530
And notice, we have no remainder.

00:03:27.530 --> 00:03:29.240
And what's interesting about
algebraic long division,

00:03:29.240 --> 00:03:31.180
and we'll probably see it
in another video or two,

00:03:31.180 --> 00:03:33.270
you can actually have a remainder.

00:03:33.270 --> 00:03:34.850
So those are going to be situations where

00:03:34.850 --> 00:03:37.520
just a factoring technique
alone would not have worked.

00:03:37.520 --> 00:03:42.170
In this situation, this
model would have been easier,

00:03:42.170 --> 00:03:43.500
but this is another way to think about it.

00:03:43.500 --> 00:03:44.430
You can say, hey, look,

00:03:44.430 --> 00:03:47.860
x plus two times x plus five
is going to be equal to this.

00:03:47.860 --> 00:03:50.730
Now, if you wanted to
rewrite this expression

00:03:50.730 --> 00:03:52.410
the way we did here, and say, hey,

00:03:52.410 --> 00:03:55.000
this expression is equal to x plus five,

00:03:55.000 --> 00:03:57.230
we would have to constrain the domain.

00:03:57.230 --> 00:04:00.320
You'd say, hey, for all x's
not equaling negative two

00:04:00.320 --> 00:04:03.353
for these to be completely
identical expressions.


Dividing quadratics by linear expressions with remainders: missing x-term

WEBVTT
Kind: captions
Language: en

00:00:00.470 --> 00:00:02.190
- [Instructor] This
polynomial division business

00:00:02.190 --> 00:00:03.890
is a little bit more fun than we expected.

00:00:03.890 --> 00:00:05.440
So let's keep going.

00:00:05.440 --> 00:00:07.580
So let's say that, I guess again,

00:00:07.580 --> 00:00:08.930
someone walks up to you in the street

00:00:08.930 --> 00:00:13.930
and says "What is x squared
plus one divided by x plus two."

00:00:14.270 --> 00:00:16.280
So pause this video and have a go at that.

00:00:16.280 --> 00:00:18.140
And I'll give you a
little bit of a warning.

00:00:18.140 --> 00:00:21.250
This one's a little bit more
involved than you might expect.

00:00:21.250 --> 00:00:23.620
All right, so there's two
ways to approach this.

00:00:23.620 --> 00:00:26.480
Either we can try to
re-express the numerator

00:00:26.480 --> 00:00:28.779
where it involves an x plus two somehow,

00:00:28.779 --> 00:00:32.100
or we could try to do
algebraic long division.

00:00:32.100 --> 00:00:33.480
So let me do the first way.

00:00:33.480 --> 00:00:35.010
So x squared plus one,

00:00:35.010 --> 00:00:38.410
it's not obvious that
you can factor it out.

00:00:38.410 --> 00:00:43.410
But can you write something
that has x plus two as a factor,

00:00:45.100 --> 00:00:47.920
and interestingly enough
has no first degree terms?

00:00:47.920 --> 00:00:50.310
'Cause we don't want
some first degree weird

00:00:50.310 --> 00:00:52.430
first degree terms sitting up there.

00:00:52.430 --> 00:00:54.610
And the best thing that
I could think of is,

00:00:54.610 --> 00:00:57.980
constructing a different of
squares using x plus two.

00:00:57.980 --> 00:01:02.980
So we know that x plus two
times x minus two is equal to

00:01:03.550 --> 00:01:06.620
x squared minus four.

00:01:06.620 --> 00:01:09.380
So what is we were to write
x squared minus four up here,

00:01:09.380 --> 00:01:13.900
and then we would just have to
add five to get to plus one.

00:01:13.900 --> 00:01:18.900
So what if we were to
write x squared minus four

00:01:19.000 --> 00:01:21.600
and then we write plus five.

00:01:21.600 --> 00:01:24.510
This expression and that
expression up there,

00:01:24.510 --> 00:01:26.730
those are completely equivalent.

00:01:26.730 --> 00:01:28.550
But why did I do that?

00:01:28.550 --> 00:01:33.380
Well, now I can write
x squared minus four as

00:01:33.380 --> 00:01:37.540
x plus two times x minus two.

00:01:37.540 --> 00:01:40.010
And so then I could rewrite
this entire expression

00:01:40.010 --> 00:01:45.010
as x plus two times x
minus two, all of that over

00:01:45.800 --> 00:01:50.800
x plus two plus five,
plus five over x plus two.

00:01:53.890 --> 00:01:58.160
And now as long as x does
not equal negative two,

00:01:58.160 --> 00:02:00.560
then we could divide the
numerator and the denominator

00:02:00.560 --> 00:02:02.500
by x plus two.

00:02:02.500 --> 00:02:07.500
And then we would be left
with x minus two plus five

00:02:08.430 --> 00:02:12.960
over x plus two, and I'll
put that little constraint,

00:02:12.960 --> 00:02:15.410
if I wanna say that this
expression is the same as that

00:02:15.410 --> 00:02:19.130
first expression, for x does not equal,

00:02:19.130 --> 00:02:21.540
for x not equaling negative two.

00:02:21.540 --> 00:02:24.860
And so here, we'd say "Hey!
X squared plus one divided by

00:02:24.860 --> 00:02:29.740
x plus two is x minus two,"
and then we have a remainder

00:02:29.740 --> 00:02:34.730
of five, remainder of five.

00:02:34.730 --> 00:02:37.750
Now let's do the same question,
or try to rewrite this

00:02:37.750 --> 00:02:39.200
using algebraic long division.

00:02:39.200 --> 00:02:40.080
We'll see that this is actually a

00:02:40.080 --> 00:02:41.820
little bit more straightforward.

00:02:41.820 --> 00:02:46.820
So we are going to divide
x plus two into x squared

00:02:49.531 --> 00:02:51.070
plus one.

00:02:51.070 --> 00:02:53.710
Now when I write things out I
like to be very careful with

00:02:53.710 --> 00:02:55.980
my, I guess you could say,
with my different places

00:02:55.980 --> 00:02:57.570
for the different degrees.

00:02:57.570 --> 00:03:00.050
So x squared plus one
has no first degree term,

00:03:00.050 --> 00:03:01.760
so I'm gonna write the one out here.

00:03:01.760 --> 00:03:05.270
So second degree, no first
degree term, and then

00:03:05.270 --> 00:03:06.170
we have a one, which is

00:03:06.170 --> 00:03:09.530
a zero degree term, or constant term.

00:03:09.530 --> 00:03:12.470
And so, we do the same
drill, how many times does

00:03:12.470 --> 00:03:13.720
x go into x squared.

00:03:13.720 --> 00:03:16.490
We're just looking at the
highest degree terms here.

00:03:16.490 --> 00:03:19.630
X goes into x squared x
times, that's first degree

00:03:19.630 --> 00:03:21.760
so I put it in the first degree column.

00:03:21.760 --> 00:03:26.260
X times two is two x.

00:03:26.260 --> 00:03:29.950
X times x is x squared.

00:03:29.950 --> 00:03:31.990
And now we wanna subtract.

00:03:31.990 --> 00:03:34.100
And so what is this gonna be equal to?

00:03:34.100 --> 00:03:36.550
We know the x squared's cancel out.

00:03:36.550 --> 00:03:38.470
And then I'm gonna be
subtracting negative two x

00:03:38.470 --> 00:03:42.570
from, you could do this
as plus zero x up here

00:03:42.570 --> 00:03:46.230
plus one, and so you're
left with negative two x.

00:03:46.230 --> 00:03:50.530
And then we bring down that one plus one.

00:03:50.530 --> 00:03:54.140
X goes into negative two
x, negative two times.

00:03:54.140 --> 00:03:56.810
Put that in the constant column.

00:03:56.810 --> 00:04:00.840
Negative two times two is negative four.

00:04:00.840 --> 00:04:05.840
And then negative two
times x is negative two x.

00:04:05.960 --> 00:04:08.050
Now we have to be very
careful here because we want

00:04:08.050 --> 00:04:10.870
to subtract the negative
two x minus four from

00:04:10.870 --> 00:04:13.230
the negative two x plus one.

00:04:13.230 --> 00:04:16.330
We could view it as this
or we could just distribute

00:04:16.330 --> 00:04:17.640
the negative sign.

00:04:17.640 --> 00:04:21.120
And then this will be
positive two x plus four.

00:04:21.120 --> 00:04:24.100
And then, the two x's, the two x

00:04:24.100 --> 00:04:25.950
and the negative two x cancels out.

00:04:25.950 --> 00:04:29.840
One plus four is five and
there's no obvious way

00:04:29.840 --> 00:04:32.520
of dividing x plus two
into five so we would call

00:04:32.520 --> 00:04:35.640
that the remainder,
exactly what we had before.

00:04:35.640 --> 00:04:37.590
When we divided with
algebraic long division,

00:04:37.590 --> 00:04:42.263
we got x minus two, x minus
two with a remainder of five.


Dividing quadratics by linear expressions with remainders

WEBVTT
Kind: captions
Language: en

00:00:00.230 --> 00:00:01.770
- [Instructor] So if you've
been watching these videos,

00:00:01.770 --> 00:00:03.490
you know that we have a lot of scenarios

00:00:03.490 --> 00:00:05.210
where people seem to be walking up to us

00:00:05.210 --> 00:00:07.619
on the street and asking
us to do math problems.

00:00:07.619 --> 00:00:09.696
And I guess this will be no different.

00:00:09.696 --> 00:00:11.670
So let's say someone walks up to you

00:00:11.670 --> 00:00:15.093
on the street and says, "Quick, you,

00:00:16.099 --> 00:00:21.099
"x squared plus five x
plus eight over x plus two,

00:00:22.177 --> 00:00:23.847
"what can this be simplified to?

00:00:23.847 --> 00:00:25.667
"Or what is x squared
plus five x plus eight

00:00:25.667 --> 00:00:27.320
"divided by x plus two?"

00:00:27.320 --> 00:00:30.630
Pause this video and see if
you can work through that.

00:00:30.630 --> 00:00:33.420
So there's two ways that
we can approach this.

00:00:33.420 --> 00:00:35.610
We can try to factor our numerator,

00:00:35.610 --> 00:00:37.440
and see if we have a common factor there,

00:00:37.440 --> 00:00:40.450
or we can try to use
algebraic long division.

00:00:40.450 --> 00:00:43.140
Let's first try to factor this numerator.

00:00:43.140 --> 00:00:45.170
And we would ideally want x plus two

00:00:45.170 --> 00:00:46.830
to be one of the factors.

00:00:46.830 --> 00:00:49.500
So let's see, what two
numbers can add up to five,

00:00:49.500 --> 00:00:51.520
and when I multiply 'em I get to eight,

00:00:51.520 --> 00:00:53.320
and ideally two is one of them.

00:00:53.320 --> 00:00:55.180
So I can think of two and three.

00:00:55.180 --> 00:01:00.180
But two times three is going
to be equal to six, not eight.

00:01:00.610 --> 00:01:02.630
And I can't think of anything else.

00:01:02.630 --> 00:01:05.470
But that still gives us some progress.

00:01:05.470 --> 00:01:07.340
Because, what if we did say,

00:01:07.340 --> 00:01:08.780
all right, let's rewrite part of it.

00:01:08.780 --> 00:01:12.180
What if we were to write
x squared plus five x,

00:01:12.180 --> 00:01:13.960
and we wanna write a plus six,

00:01:13.960 --> 00:01:17.770
because that actually would
be divisible by x plus two.

00:01:17.770 --> 00:01:19.900
So I'm gonna write a plus six.

00:01:19.900 --> 00:01:21.230
But of course we have an eight here,

00:01:21.230 --> 00:01:24.060
so then we're going to have
an extra two right over there.

00:01:24.060 --> 00:01:28.630
And then all of that is
divisible by x plus two.

00:01:28.630 --> 00:01:33.630
And now I can rewrite this
part, up here in orange.

00:01:34.020 --> 00:01:36.930
That is x plus two times x plus three.

00:01:36.930 --> 00:01:37.990
So let me write it here.

00:01:37.990 --> 00:01:42.310
x plus two times x plus three,

00:01:42.310 --> 00:01:44.430
I still have that plus
two sitting out there

00:01:44.430 --> 00:01:47.010
in the numerator, plus two.

00:01:47.010 --> 00:01:49.990
And then all of that over x plus two.

00:01:49.990 --> 00:01:54.260
Or I could write this as
being over x plus two.

00:01:54.260 --> 00:01:58.860
And this being over x plus two.

00:01:58.860 --> 00:02:00.120
All I did is I said, hey,

00:02:00.120 --> 00:02:02.029
if I have something plus something else

00:02:02.029 --> 00:02:04.610
over x plus two, that could
be the first something

00:02:04.610 --> 00:02:06.640
over x plus two plus the second something

00:02:06.640 --> 00:02:08.140
over x plus two.

00:02:08.140 --> 00:02:10.730
And then here, we can say, hey, look,

00:02:10.730 --> 00:02:14.880
this first part, as long as x
does not equal negative two,

00:02:14.880 --> 00:02:17.720
because then we would
be changing the domain,

00:02:17.720 --> 00:02:20.420
then these two would cancel out.

00:02:20.420 --> 00:02:21.690
What you could say, hey, I'm just dividing

00:02:21.690 --> 00:02:24.010
the numerator and the
denominator by x plus two.

00:02:24.010 --> 00:02:29.010
And so this would be equal
to x plus three, plus,

00:02:29.430 --> 00:02:31.980
and I don't necessarily even
have to put parentheses there,

00:02:31.980 --> 00:02:36.980
plus two over x plus two.

00:02:39.150 --> 00:02:43.290
And I would have to constrain the domain

00:02:43.290 --> 00:02:46.960
so this is for x does
not equal negative two.

00:02:46.960 --> 00:02:50.280
So in this situation, we had a remainder.

00:02:50.280 --> 00:02:54.160
And people will refer to
the two as the remainder.

00:02:54.160 --> 00:02:55.900
We divide it as far as we can,

00:02:55.900 --> 00:02:58.360
but it still remains to be done

00:02:58.360 --> 00:03:00.830
to divide the two by x plus two.

00:03:00.830 --> 00:03:05.740
And so we would refer to
the two as the remainder.

00:03:05.740 --> 00:03:08.235
Now that wasn't too difficult, but it also

00:03:08.235 --> 00:03:11.050
wasn't too straightforward and we'll see

00:03:11.050 --> 00:03:14.320
that this is a situation where
the algebraic long division

00:03:14.320 --> 00:03:16.300
is actually a little bit
more straightforward.

00:03:16.300 --> 00:03:17.750
So let's try that out.

00:03:17.750 --> 00:03:19.112
Once again, pause this
video and see if you can

00:03:19.112 --> 00:03:22.490
figure out what this is through
algebraic long division.

00:03:22.490 --> 00:03:26.830
So we're trying to take x
plus two and divide it into

00:03:28.040 --> 00:03:32.270
x squared plus five x plus eight.

00:03:32.270 --> 00:03:35.890
Look at the highest degree
terms, the x and the x squared.

00:03:35.890 --> 00:03:38.630
x goes into x squared x times,

00:03:38.630 --> 00:03:40.950
put it in the first degree column.

00:03:40.950 --> 00:03:45.840
x times two is two x,
x times x is x squared,

00:03:45.840 --> 00:03:49.860
subtract these from x squared plus five x

00:03:49.860 --> 00:03:53.100
and we get five x minus two x is three x.

00:03:53.100 --> 00:03:56.390
x squared minus x
squared, that's just zero.

00:03:56.390 --> 00:03:58.767
Bring down that eight.

00:03:58.767 --> 00:04:01.330
Look at the highest-degree term.

00:04:01.330 --> 00:04:04.740
And we get x goes into
three x three times.

00:04:04.740 --> 00:04:06.410
Put that in the constant column,

00:04:06.410 --> 00:04:09.960
or the zeroth degree
column, so plus three.

00:04:09.960 --> 00:04:14.560
Three times two is six,
three times x is three x.

00:04:14.560 --> 00:04:17.720
Subtract ds, and we are left with,

00:04:17.720 --> 00:04:19.970
let me scroll down a little bit,

00:04:19.970 --> 00:04:21.930
you're left with, those cancel out,

00:04:21.930 --> 00:04:24.340
and you're left with eight minus six,

00:04:24.340 --> 00:04:27.990
which is, indeed, equal to two.

00:04:27.990 --> 00:04:30.300
And we could say, hey, we
don't really know how to divide

00:04:30.300 --> 00:04:33.610
x plus two into two for an arbitrary x,

00:04:33.610 --> 00:04:36.160
so we will say, hey, this
is going to be equal to

00:04:36.160 --> 00:04:40.467
x plus three with a remainder of two.

00:04:43.890 --> 00:04:46.470
Now once again, if you wanted
to rewrite that original

00:04:46.470 --> 00:04:48.690
expression, and you wanted
it to be completely the same,

00:04:48.690 --> 00:04:51.860
including the domain, you
would have to constrain,

00:04:51.860 --> 00:04:53.670
you would have to constrain the domain,

00:04:53.670 --> 00:04:54.713
just like that.


Simple polynomial division

WEBVTT
Kind: captions
Language: en

00:00:00.200 --> 00:00:01.770
- [Instructor] Let's say
someone walks up to you

00:00:01.770 --> 00:00:03.710
on the street, and they
give you this expression,

00:00:03.710 --> 00:00:06.990
x squared plus seven x plus 10

00:00:06.990 --> 00:00:10.300
divided by x plus two.

00:00:10.300 --> 00:00:13.550
And they say, "See if you
could simplify this thing."

00:00:13.550 --> 00:00:16.610
And so pause this video,
and see if you can do that.

00:00:16.610 --> 00:00:18.972
And one way to think about it is what is

00:00:18.972 --> 00:00:22.450
x squared plus seven x plus
10 divided by x plus two?

00:00:22.450 --> 00:00:23.700
What is that going to be?

00:00:24.860 --> 00:00:27.330
All right, now there's two ways
that you can approach this.

00:00:27.330 --> 00:00:29.800
One way is to try to factor the numerator

00:00:29.800 --> 00:00:31.770
and see if it has a factor

00:00:31.770 --> 00:00:34.390
that is common to the denominator.

00:00:34.390 --> 00:00:35.700
So let's try to do that.

00:00:35.700 --> 00:00:37.540
So we've done this many, many times.

00:00:37.540 --> 00:00:39.990
If this looks new to you,
I encourage you to review

00:00:39.990 --> 00:00:42.950
factoring polynomials other
places on Khan Academy.

00:00:42.950 --> 00:00:44.900
But what two numbers add up to seven

00:00:44.900 --> 00:00:47.180
and when you multiply 'em you get 10?

00:00:47.180 --> 00:00:48.790
Well, that would be two and five.

00:00:48.790 --> 00:00:52.240
So we could rewrite that
numerator as x plus two

00:00:52.240 --> 00:00:54.600
times x plus five.

00:00:54.600 --> 00:00:56.070
And then, of course, the denominator,

00:00:56.070 --> 00:00:59.090
you still have x plus two.

00:00:59.090 --> 00:01:01.800
And then we clearly see
we have a common factor.

00:01:01.800 --> 00:01:05.650
And so as long as x does
not equal negative two,

00:01:05.650 --> 00:01:07.310
because if x equals negative two,

00:01:07.310 --> 00:01:09.170
this whole expression is undefined

00:01:09.170 --> 00:01:11.140
because then you get a
zero in the denominator.

00:01:11.140 --> 00:01:15.430
So as long as x does
not equal negative two,

00:01:15.430 --> 00:01:17.390
well, then we can divide the numerator

00:01:17.390 --> 00:01:19.840
and the denominator by x plus two.

00:01:19.840 --> 00:01:21.470
Once again, the reason why
I put that constraint is

00:01:21.470 --> 00:01:23.920
we can't divide the numerator
and the denominator by zero.

00:01:23.920 --> 00:01:25.920
So for any other values of x,

00:01:25.920 --> 00:01:28.950
this x plus two will be
nonzero, and we could divide

00:01:28.950 --> 00:01:30.910
the numerator and the denominator by that.

00:01:30.910 --> 00:01:32.180
And they would cancel out,

00:01:32.180 --> 00:01:36.540
and we would just be
left with x plus five.

00:01:36.540 --> 00:01:39.000
So another way to think
about it is this expression,

00:01:39.000 --> 00:01:42.230
our original expression,
could be viewed as x plus five

00:01:42.230 --> 00:01:47.230
for any x that is not
equal to negative two.

00:01:47.430 --> 00:01:50.060
Now, the other way that
we could approach this

00:01:50.060 --> 00:01:53.310
is through algebraic long division,

00:01:53.310 --> 00:01:55.640
which is very analogous to
the type of long division

00:01:55.640 --> 00:01:59.060
that you might remember from,
I believe it was fourth grade.

00:01:59.060 --> 00:02:00.130
So what you do is you say,

00:02:00.130 --> 00:02:03.500
all right, I'm going to
divide x plus two into

00:02:03.500 --> 00:02:07.830
x squared plus seven x plus 10.

00:02:07.830 --> 00:02:10.740
And in this technique, you look
at the highest-degree term.

00:02:10.740 --> 00:02:13.620
So you have an x there
and an x squared there.

00:02:13.620 --> 00:02:16.310
You say, how many times
does x go into x squared?

00:02:16.310 --> 00:02:18.120
Well, it goes x times.

00:02:18.120 --> 00:02:20.070
Now, you'd write that in this column

00:02:20.070 --> 00:02:22.830
because x is just x to the first power.

00:02:22.830 --> 00:02:24.950
You could view this as
the first degree column.

00:02:24.950 --> 00:02:27.890
It's analogous to the place values

00:02:27.890 --> 00:02:30.570
that we talk about when
we first learn numbers

00:02:30.570 --> 00:02:33.790
or how we regroup or about place value.

00:02:33.790 --> 00:02:35.720
But here, you could
view it as degree places

00:02:35.720 --> 00:02:36.740
or something like that.

00:02:36.740 --> 00:02:37.960
And then you take that x,

00:02:37.960 --> 00:02:40.360
and you multiply it times
this entire expression.

00:02:40.360 --> 00:02:42.930
So x times two is two x.

00:02:42.930 --> 00:02:44.860
Put that in the first degree column.

00:02:44.860 --> 00:02:48.440
X times x is x squared.

00:02:48.440 --> 00:02:50.850
And then what we want to
do is we want to subtract

00:02:50.850 --> 00:02:54.310
these things in yellow from
what we originally had in blue.

00:02:54.310 --> 00:02:55.960
So we could do it this way.

00:02:55.960 --> 00:02:57.910
And then we will be left with

00:02:57.910 --> 00:03:00.620
seven x minus two x is five x.

00:03:00.620 --> 00:03:03.930
And then x squared minus
x squared is just zero.

00:03:03.930 --> 00:03:08.330
And then we can bring down this plus 10.

00:03:08.330 --> 00:03:11.080
And once again, we look at
the highest degree term.

00:03:11.080 --> 00:03:13.730
X goes into five x five times.

00:03:13.730 --> 00:03:14.660
That's a zero degree.

00:03:14.660 --> 00:03:17.750
It's a constant, so I'll write
it in the constant column.

00:03:17.750 --> 00:03:19.990
Five times two is 10.

00:03:19.990 --> 00:03:22.430
Five times x is five.

00:03:22.430 --> 00:03:25.520
And then I'll subtract these
from what we have up here.

00:03:25.520 --> 00:03:27.540
And notice, we have no remainder.

00:03:27.540 --> 00:03:29.270
What's interesting about
algebraic long division,

00:03:29.270 --> 00:03:31.180
we'll probably see it
in another video or two,

00:03:31.180 --> 00:03:33.270
you can actually have a remainder.

00:03:33.270 --> 00:03:35.100
So those are going to
be situations where just

00:03:35.100 --> 00:03:37.520
the factoring technique
alone would not have worked.

00:03:37.520 --> 00:03:39.570
In this situation, this,

00:03:39.570 --> 00:03:42.170
this model would have been easier,

00:03:42.170 --> 00:03:43.500
but this is another way to think about it.

00:03:43.500 --> 00:03:45.860
You can say, hey, look, x
plus two times x plus five

00:03:45.860 --> 00:03:47.860
is going to be equal to this.

00:03:47.860 --> 00:03:50.730
Now, if you wanted to
rewrite this expression

00:03:50.730 --> 00:03:52.110
the way we did here and say,

00:03:52.110 --> 00:03:55.000
hey, this expression is
equal to x plus five,

00:03:55.000 --> 00:03:57.230
we would have to constrain the domain.

00:03:57.230 --> 00:04:00.320
You'd say, hey, for all x's
not equaling negative two,

00:04:00.320 --> 00:04:03.353
for these to be completely
identical expressions.


Dividing polynomials by linear expressions

WEBVTT
Kind: captions
Language: en

00:00:00.240 --> 00:00:02.900
- We're told, divide the polynomials.

00:00:02.900 --> 00:00:05.230
The form of your answer should either be

00:00:05.230 --> 00:00:06.620
just a clean polynomial,

00:00:06.620 --> 00:00:10.470
or some polynomial plus some
constant over x plus two,

00:00:10.470 --> 00:00:13.310
where p of x is a polynomial
and k is an integer.

00:00:13.310 --> 00:00:15.700
Fair enough, and if we were
doing this on Khan Academy,

00:00:15.700 --> 00:00:17.700
this is a screen shot from Khan Academy,

00:00:17.700 --> 00:00:18.820
we would have to type this in,

00:00:18.820 --> 00:00:20.220
but we're just going to do it by hand.

00:00:20.220 --> 00:00:22.290
And, like always, pause this video

00:00:22.290 --> 00:00:23.470
and try to do it on your own

00:00:23.470 --> 00:00:26.053
before we work through it together.

00:00:26.960 --> 00:00:28.490
All right, now let's
work through it together.

00:00:28.490 --> 00:00:32.890
And what we're trying to
do is divide x plus two,

00:00:32.890 --> 00:00:37.340
into three x to the third power

00:00:37.340 --> 00:00:42.290
plus four x squared
minus three x plus seven.

00:00:42.290 --> 00:00:43.940
And so, like always,

00:00:43.940 --> 00:00:47.650
we focus on the highest
degree terms first.

00:00:47.650 --> 00:00:50.470
x goes into three x to the
third power how many times?

00:00:50.470 --> 00:00:52.070
Three x squared times.

00:00:52.070 --> 00:00:55.200
We'd want to put that in
the second degree column.

00:00:55.200 --> 00:00:56.940
Three x squared.

00:00:56.940 --> 00:01:00.370
Three x squared times
two is six x squared.

00:01:00.370 --> 00:01:04.900
Three x squared times x is
three x to the third power.

00:01:04.900 --> 00:01:06.130
There's something very meditative

00:01:06.130 --> 00:01:07.870
about algebraic long division.

00:01:07.870 --> 00:01:10.180
Anyway, we'd want to
subtract what we just wrote

00:01:10.180 --> 00:01:11.750
from what we have up here.

00:01:11.750 --> 00:01:13.260
So let's subtract.

00:01:13.260 --> 00:01:15.810
And these characters cancel out.

00:01:15.810 --> 00:01:18.650
And then four x squared
minus six x squared

00:01:18.650 --> 00:01:21.620
is negative two x squared.

00:01:21.620 --> 00:01:24.510
Bring down that negative three x.

00:01:24.510 --> 00:01:26.410
And now we would wanna say,

00:01:26.410 --> 00:01:30.960
hey, how many times does x go
into negative two x squared?

00:01:30.960 --> 00:01:33.500
Well, it would go negative two x times.

00:01:33.500 --> 00:01:35.660
Put that in our first degree column.

00:01:35.660 --> 00:01:39.900
Negative two x times
two is negative four x.

00:01:39.900 --> 00:01:44.470
Negative two x times x is
negative two x squared.

00:01:44.470 --> 00:01:46.750
Now we wanna subtract what
we have here in orange

00:01:46.750 --> 00:01:48.940
from what we have up here in teal.

00:01:48.940 --> 00:01:51.940
So we either put a negative
around the whole thing

00:01:51.940 --> 00:01:54.680
or we distribute that negative
and that becomes a positive,

00:01:54.680 --> 00:01:56.560
that becomes a positive.

00:01:56.560 --> 00:02:00.000
And so this is equal to, the
x squared terms cancel out.

00:02:00.000 --> 00:02:01.930
Negative three x plus four x

00:02:01.930 --> 00:02:04.260
is just going to be a straight up x.

00:02:04.260 --> 00:02:07.920
Bring down that seven, x plus seven.

00:02:07.920 --> 00:02:10.980
How many times does x go into x?

00:02:10.980 --> 00:02:12.360
Well, one time.

00:02:12.360 --> 00:02:14.580
Actually, let me do a,
use a new color here.

00:02:14.580 --> 00:02:16.910
So, how many times does x go into x?

00:02:16.910 --> 00:02:18.540
It goes one time.

00:02:18.540 --> 00:02:20.520
Put that in the constant column.

00:02:20.520 --> 00:02:22.270
One times two is two.

00:02:22.270 --> 00:02:23.770
One times x is x.

00:02:23.770 --> 00:02:25.730
We wanna subtract these characters.

00:02:25.730 --> 00:02:30.730
And we're left with
seven minus two is five.

00:02:30.900 --> 00:02:35.900
And so, we can rewrite
this whole thing as,

00:02:36.160 --> 00:02:39.120
we deserve I guess a
little bit of a drum roll.

00:02:39.120 --> 00:02:43.640
Three x squared minus two x plus one,

00:02:43.640 --> 00:02:46.360
plus the remainder,

00:02:46.360 --> 00:02:49.140
five, over x plus two.

00:02:49.140 --> 00:02:50.210
One way to think about it is,

00:02:50.210 --> 00:02:51.220
hey, I have this remainder,

00:02:51.220 --> 00:02:53.790
I'd have to keep dividing it by x plus two

00:02:53.790 --> 00:02:56.510
if I really wanted to figure
out exactly what this is.

00:02:56.510 --> 00:02:59.720
Now if I wanted these expressions
to be completely identical

00:02:59.720 --> 00:03:02.820
I would put a condition on the domain

00:03:02.820 --> 00:03:05.340
that x cannot be equal to negative two

00:03:05.340 --> 00:03:07.030
because if x was equal to negative two,

00:03:07.030 --> 00:03:09.060
we'd be dividing by zero here.

00:03:09.060 --> 00:03:11.210
But for the purposes of this exercise,

00:03:11.210 --> 00:03:16.140
you just have to input
this part right over here,

00:03:16.140 --> 00:03:17.400
you'd have to type it in,

00:03:17.400 --> 00:03:19.730
which I guess isn't the easiest
thing to do in the world.

00:03:19.730 --> 00:03:21.630
But it's worth doing.

00:03:21.630 --> 00:03:23.480
All right, see you in the next video.


Dividing polynomials by linear expressions: missing term

WEBVTT
Kind: captions
Language: en

00:00:00.270 --> 00:00:02.680
In front of us, we have another screenshot

00:00:02.680 --> 00:00:04.720
from Khan Academy and I've
modified it a little bit

00:00:04.720 --> 00:00:06.170
so I have a little bit of extra space.

00:00:06.170 --> 00:00:08.490
And it says, divide the polynomials.

00:00:08.490 --> 00:00:10.099
The form of your answer should either be

00:00:10.099 --> 00:00:12.176
a straight up polynomial or a polynomial

00:00:12.176 --> 00:00:15.610
plus the remainder over x minus five,

00:00:15.610 --> 00:00:17.934
which we have here in the denominator,

00:00:17.934 --> 00:00:20.010
where p of x is a polynomial
and k is an integer.

00:00:20.010 --> 00:00:21.310
So, we've done stuff like this,

00:00:21.310 --> 00:00:23.950
but like always, I encourage
you to pause this video

00:00:23.950 --> 00:00:25.460
and work on this on your own.

00:00:25.460 --> 00:00:27.485
And if you were doing
this on Khan Academy,

00:00:27.485 --> 00:00:29.080
there's a little bit of an input box here

00:00:29.080 --> 00:00:29.913
where you'd have to type in the answer,

00:00:29.913 --> 00:00:31.810
but let's just do it on paper for now.

00:00:32.740 --> 00:00:33.660
All right.

00:00:33.660 --> 00:00:34.900
So, we're trying to figure out

00:00:34.900 --> 00:00:39.490
what x minus five divided into

00:00:41.080 --> 00:00:43.460
two x to the third power.

00:00:43.460 --> 00:00:44.840
Actually, I want to be careful here

00:00:44.840 --> 00:00:48.270
because I want to be very, very organized

00:00:48.270 --> 00:00:50.630
about my different degree columns.

00:00:50.630 --> 00:00:52.440
So, this is my third degree column.

00:00:52.440 --> 00:00:54.550
And then I want my second degree column.

00:00:54.550 --> 00:00:56.670
But there is no second degree term here.

00:00:56.670 --> 00:00:57.730
There's a first degree term.

00:00:57.730 --> 00:00:59.080
So, I'll write it out here.

00:01:00.112 --> 00:01:00.945
So, minus 47x.

00:01:02.420 --> 00:01:03.720
And actually, to be even more careful

00:01:03.720 --> 00:01:06.930
I'll write plus zero x squared.

00:01:06.930 --> 00:01:09.226
And then I have minus 15.

00:01:09.226 --> 00:01:11.930
By putting that plus zero x squared,

00:01:11.930 --> 00:01:14.290
that's making sure I'm doing good,

00:01:14.290 --> 00:01:17.273
I guess degree place column hygiene.

00:01:17.273 --> 00:01:20.217
All right, so now we
can work through this.

00:01:20.217 --> 00:01:23.200
And first we could say hey, how many times

00:01:23.200 --> 00:01:25.550
does x go into the
highest degree term here?

00:01:25.550 --> 00:01:27.520
Well, x goes into two x to the third power

00:01:27.520 --> 00:01:29.880
two x squared times.

00:01:29.880 --> 00:01:32.730
We'd want to put that in
the second degree column.

00:01:32.730 --> 00:01:33.990
Two x squared.

00:01:33.990 --> 00:01:35.500
You can see how it would've gotten messy

00:01:35.500 --> 00:01:37.470
if I'd put negative 47x here.

00:01:37.470 --> 00:01:39.450
I'd be like where do I
put that two x squared

00:01:39.450 --> 00:01:41.870
and you might confused yourself,

00:01:41.870 --> 00:01:44.340
which none of us would want to happen.

00:01:44.340 --> 00:01:46.990
All right, two x squared
times negative five

00:01:46.990 --> 00:01:50.330
is negative 10x squared.

00:01:50.330 --> 00:01:54.039
Two x squared times x is
2x to the third power.

00:01:54.039 --> 00:01:56.310
Now we want to subtract
what we have in red

00:01:56.310 --> 00:01:57.830
from what we have in blue.

00:01:57.830 --> 00:02:00.390
So, I'll multiply them
both by negative one.

00:02:00.390 --> 00:02:01.580
So that becomes a negative

00:02:01.580 --> 00:02:03.740
and then that one becomes a positive.

00:02:03.740 --> 00:02:05.460
And that's actually one
of the biggest areas

00:02:05.460 --> 00:02:06.660
for careless errors.

00:02:06.660 --> 00:02:09.330
If you had negative here and
you just want to subtract it

00:02:09.330 --> 00:02:11.120
because you know you have to subtract it.

00:02:11.120 --> 00:02:12.859
Be like no, I'm subtracting a negative,

00:02:12.859 --> 00:02:13.790
it needs to be a positive now.

00:02:13.790 --> 00:02:14.950
All right.

00:02:14.950 --> 00:02:19.730
So, zero x squared plus
10x squared is 10x squared.

00:02:19.730 --> 00:02:21.610
And then the two x to the third

00:02:21.610 --> 00:02:24.000
minus two x to the third is just zero.

00:02:24.000 --> 00:02:27.273
And then we can bring
down that negative 47x.

00:02:29.330 --> 00:02:34.317
And once again, we look at
the highest degree terms.

00:02:34.317 --> 00:02:38.775
X goes into 10x squared 10x times.

00:02:38.775 --> 00:02:41.480
So, plus 10x.

00:02:41.480 --> 00:02:45.673
10x times negative five is negative 50x.

00:02:48.590 --> 00:02:52.270
10x times x is 10x squared.

00:02:52.270 --> 00:02:55.220
And once again, we want to
subtract what we have in teal

00:02:55.220 --> 00:02:56.600
from what we have in red.

00:02:56.600 --> 00:02:59.080
So we could multiply both
of these times negative one.

00:02:59.080 --> 00:03:00.480
That becomes a negative.

00:03:00.480 --> 00:03:02.373
This one becomes a positive.

00:03:03.240 --> 00:03:08.230
Now, negative 47x plus
50x is positive three x.

00:03:08.230 --> 00:03:12.000
And then 10x squared minus
10x squared gets canceled out.

00:03:12.000 --> 00:03:14.340
Bring down that 15.

00:03:14.340 --> 00:03:16.030
Come on down.

00:03:16.030 --> 00:03:18.810
Used to watch a lot of
Price is Right growing up.

00:03:18.810 --> 00:03:20.490
Never quite made it to the show.

00:03:20.490 --> 00:03:21.660
All right.

00:03:21.660 --> 00:03:25.260
X goes into three x how many times?

00:03:25.260 --> 00:03:27.550
It goes three times.

00:03:27.550 --> 00:03:30.940
Three times negative five is negative 15.

00:03:30.940 --> 00:03:33.360
Three times x is three x.

00:03:33.360 --> 00:03:36.320
We want to subtract the
orange from the teal.

00:03:36.320 --> 00:03:40.290
And so, this becomes a negative,
this becomes a positive.

00:03:40.290 --> 00:03:43.610
Negative 15 plus 15 is zero.

00:03:43.610 --> 00:03:46.213
And three x minus three x is zero.

00:03:46.213 --> 00:03:47.594
So, you're just left with zero.

00:03:47.594 --> 00:03:48.813
So, no remainder.

00:03:48.813 --> 00:03:53.100
So this whole thing, you could
re-express or simplify as

00:03:53.100 --> 00:03:57.100
two x squared plus 10x plus three.

00:03:57.100 --> 00:03:58.470
And once again, if this
was on Khan Academy,

00:03:58.470 --> 00:03:59.900
there would be a little
bit of an input box

00:03:59.900 --> 00:04:01.682
that looks something like this

00:04:01.682 --> 00:04:03.500
and you would have to type this in.

00:04:03.500 --> 00:04:06.580
Now, if you wanted these to
be exactly the same expression

00:04:06.580 --> 00:04:08.690
you would also need to
constrain the domain.

00:04:08.690 --> 00:04:12.860
You would say, okay four x
does not equal positive five.

00:04:12.860 --> 00:04:14.190
And the reason why you
have to constrain that

00:04:14.190 --> 00:04:18.300
is the whole reason why we can
even divide by x minus five

00:04:18.300 --> 00:04:21.510
is we're assuming that x minus
five is no equal to zero.

00:04:21.510 --> 00:04:23.280
And it's generally not equal to zero

00:04:23.280 --> 00:04:26.410
as long as x does not
equal to positive five.

00:04:26.410 --> 00:04:28.120
But for the sake of that exercise,

00:04:28.120 --> 00:04:30.863
you don't need to constrain
the domain like this.


Factoring using polynomial division: missing term

WEBVTT
Kind: captions
Language: en

00:00:00.150 --> 00:00:02.760
- [Instructor] We're told
the polynomial p of x,

00:00:02.760 --> 00:00:06.840
which is equal to this, has
a known factor of x plus six.

00:00:06.840 --> 00:00:10.470
Rewrite p of x as a
product of linear factors.

00:00:10.470 --> 00:00:13.290
Pause this video, and see if
you can have a go at that.

00:00:13.290 --> 00:00:16.040
All right, now let's
work on this together.

00:00:16.040 --> 00:00:18.030
Because they give us one of the factors,

00:00:18.030 --> 00:00:19.340
what we can do is say hey,

00:00:19.340 --> 00:00:22.970
what happens if I divide
x plus six into p of x?

00:00:22.970 --> 00:00:24.200
What do I have left over?

00:00:24.200 --> 00:00:26.520
It looks like I'm still
going to have a quadratic,

00:00:26.520 --> 00:00:28.590
and then I'll probably
have to factor that somehow

00:00:28.590 --> 00:00:30.440
to get a product of linear factors.

00:00:30.440 --> 00:00:32.320
So let's get going.

00:00:32.320 --> 00:00:33.960
So if I were to try to figure out

00:00:33.960 --> 00:00:38.370
what x plus six divided into,

00:00:38.370 --> 00:00:42.240
x to the third plus nine x squared,

00:00:42.240 --> 00:00:43.710
and now we're gonna have to be careful.

00:00:43.710 --> 00:00:46.860
You might be tempted to
just write minus 108 there,

00:00:46.860 --> 00:00:48.250
but then this gets tricky

00:00:48.250 --> 00:00:49.930
because you have your third degree column,

00:00:49.930 --> 00:00:51.030
your second degree column.

00:00:51.030 --> 00:00:52.420
You need your first degree column,

00:00:52.420 --> 00:00:53.800
but you just put your zero degree,

00:00:53.800 --> 00:00:55.500
your constant column here.

00:00:55.500 --> 00:00:57.570
So to make sure we have good hygiene,

00:00:57.570 --> 00:01:00.200
we could write plus zero x,

00:01:00.200 --> 00:01:02.600
and I encourage you to
actually always do this

00:01:02.600 --> 00:01:04.550
if you're writing out a polynomial

00:01:04.550 --> 00:01:08.260
so that you don't skip
that place, so to speak,

00:01:08.260 --> 00:01:11.130
minus 108.

00:01:11.130 --> 00:01:13.280
And so then you say, all right,

00:01:13.280 --> 00:01:15.510
let's look at the highest degree terms.

00:01:15.510 --> 00:01:19.970
X goes into x to the
third x squared times.

00:01:19.970 --> 00:01:22.900
X squared times six is six x squared.

00:01:22.900 --> 00:01:25.180
X squared times x is x to the third.

00:01:25.180 --> 00:01:26.570
We want to subtract.

00:01:26.570 --> 00:01:27.770
We've done this multiple times,

00:01:27.770 --> 00:01:30.000
so I'm going a little
bit faster than normal.

00:01:30.000 --> 00:01:30.960
Those cancel out.

00:01:30.960 --> 00:01:35.020
Nine x squared minus six x
squared is three x squared.

00:01:35.020 --> 00:01:37.183
Bring down that zero x.

00:01:38.470 --> 00:01:42.130
And then how many times does
x go into three x squared?

00:01:42.130 --> 00:01:43.810
Well, it goes three x times,

00:01:43.810 --> 00:01:45.700
and we would write it in this column.

00:01:45.700 --> 00:01:48.210
And notice, if we didn't keep this column

00:01:48.210 --> 00:01:50.390
for our first degree terms,
we'd be kind of confused

00:01:50.390 --> 00:01:52.540
where to write that
three x right about now.

00:01:52.540 --> 00:01:57.450
And so three x plus, times
six, I should say, is 18 x.

00:01:57.450 --> 00:01:59.930
Three x times x is three x squared.

00:01:59.930 --> 00:02:01.750
We want to subtract what we have in that,

00:02:01.750 --> 00:02:05.120
I guess that color is mauve,
light purple, not sure.

00:02:05.120 --> 00:02:09.000
And so we get three x squareds cancel out,

00:02:09.000 --> 00:02:12.140
and then zero x minus 18 x

00:02:12.140 --> 00:02:14.420
is negative 18 x.

00:02:14.420 --> 00:02:18.500
Bring down that negative 108.

00:02:18.500 --> 00:02:21.180
And so then we have

00:02:21.180 --> 00:02:24.250
x goes into negative 18 x

00:02:24.250 --> 00:02:26.870
negative 18 times.

00:02:26.870 --> 00:02:30.690
Negative 18 times six is

00:02:30.690 --> 00:02:32.750
negative 108.

00:02:32.750 --> 00:02:34.300
That's working out nicely.

00:02:34.300 --> 00:02:37.430
Negative 18 times x is negative 18 x.

00:02:37.430 --> 00:02:38.820
And then we want to subtract

00:02:38.820 --> 00:02:41.120
what we have in this
not-so-pleasant brown color,

00:02:41.120 --> 00:02:44.280
and so I will multiply
them both by a negative.

00:02:44.280 --> 00:02:46.930
And so I am left with zero.

00:02:46.930 --> 00:02:48.790
Everything just cancels out.

00:02:48.790 --> 00:02:52.000
And so I can rewrite p of x.

00:02:52.000 --> 00:02:54.470
I can rewrite p of x

00:02:54.470 --> 00:02:57.900
as being equal to x plus six

00:02:57.900 --> 00:03:02.170
times x squared plus three x minus 18.

00:03:02.170 --> 00:03:04.990
But I'm not done yet because
this is not a linear factor.

00:03:04.990 --> 00:03:06.570
This is still quadratic.

00:03:06.570 --> 00:03:08.160
So let's see, can I think of two numbers

00:03:08.160 --> 00:03:10.000
that add up to three

00:03:10.000 --> 00:03:12.290
and that when I multiply
I get negative 18?

00:03:12.290 --> 00:03:14.000
So I'll need different signs,

00:03:14.000 --> 00:03:16.390
and then the obvious one

00:03:18.003 --> 00:03:21.950
is positive six and negative three.

00:03:21.950 --> 00:03:24.800
And if that, what I just did,
seems like voodoo to you,

00:03:24.800 --> 00:03:28.170
I encourage you to review
factoring polynomials.

00:03:28.170 --> 00:03:32.540
But this I can rewrite
'cause negative six plus,

00:03:32.540 --> 00:03:35.030
or actually I should say positive six

00:03:35.030 --> 00:03:37.490
plus negative three is equal to three,

00:03:37.490 --> 00:03:39.380
and then positive six times negative three

00:03:39.380 --> 00:03:40.900
is equal to negative 18.

00:03:40.900 --> 00:03:45.020
So I can rewrite this as x plus six times

00:03:45.020 --> 00:03:48.620
x plus six times

00:03:48.620 --> 00:03:51.420
x minus three.

00:03:51.420 --> 00:03:52.780
And so there we have it.

00:03:52.780 --> 00:03:55.770
We have a product of linear factors.

00:03:55.770 --> 00:03:56.883
And we are done.


Geometric series word problems: swing

WEBVTT
Kind: captions
Language: en

00:00:00.330 --> 00:00:03.100
- [Instructor] We're told a
monkey is swinging from a tree.

00:00:03.100 --> 00:00:08.100
On the first swing, she passes
through an arc of 24 meters.

00:00:08.180 --> 00:00:10.930
With each swing, she passes through an arc

00:00:10.930 --> 00:00:14.120
1/2 the length of the previous swing.

00:00:14.120 --> 00:00:15.050
So what's going on here?

00:00:15.050 --> 00:00:16.530
Let's say this is the top of the rope

00:00:16.530 --> 00:00:18.800
or the vine that the
monkey is swinging from.

00:00:18.800 --> 00:00:21.140
And so on that first swing,

00:00:21.140 --> 00:00:22.870
I like to draw a little monkey here,

00:00:22.870 --> 00:00:25.690
so this is my little monkey.

00:00:25.690 --> 00:00:30.690
So on the first swing, the
monkey will go 24 meters.

00:00:30.740 --> 00:00:33.250
Might do something like this.

00:00:33.250 --> 00:00:35.030
Then that arc is 24 meters,

00:00:35.030 --> 00:00:36.700
and then on the second swing,

00:00:36.700 --> 00:00:38.810
it would be, she'd swing back at an arc

00:00:38.810 --> 00:00:41.810
half the length of the previous swing.

00:00:41.810 --> 00:00:43.650
So then she would come back

00:00:43.650 --> 00:00:45.610
and then it would be half the length,

00:00:45.610 --> 00:00:47.510
and so maybe swing back over here.

00:00:47.510 --> 00:00:49.650
And then on the next, so that'd be 12,

00:00:49.650 --> 00:00:53.370
and then on the next swing,
she would swing half of that,

00:00:53.370 --> 00:00:55.350
which would be six meters.

00:00:55.350 --> 00:00:56.980
And so she might swing like this,

00:00:56.980 --> 00:00:57.840
and that makes sense.

00:00:57.840 --> 00:01:00.630
That's consistent with our
experiences swinging from trees,

00:01:00.630 --> 00:01:02.410
for those of us who have done that.

00:01:02.410 --> 00:01:04.740
(laughs) So let's look
at the first choice.

00:01:04.740 --> 00:01:06.840
Which expression gives the total length

00:01:06.840 --> 00:01:10.980
the monkey swings in her first n swings?

00:01:10.980 --> 00:01:12.640
So pause the video and
see if you can do that,

00:01:12.640 --> 00:01:14.350
and you can express it as,

00:01:14.350 --> 00:01:15.710
actually express it two ways,

00:01:15.710 --> 00:01:18.440
express it as a geometric series,

00:01:18.440 --> 00:01:22.210
but also express it as the
sum of a geometric series

00:01:22.210 --> 00:01:23.773
if it were actually evaluated.

00:01:25.130 --> 00:01:26.790
So let's do this together.

00:01:26.790 --> 00:01:29.390
So we already said on the first swing,

00:01:29.390 --> 00:01:33.210
the monkey goes 24 meters.

00:01:33.210 --> 00:01:35.000
Now on the second swing,
and I gave you a hint

00:01:35.000 --> 00:01:38.360
when I said to express
it as a geometric series,

00:01:38.360 --> 00:01:39.980
she swings half that.

00:01:39.980 --> 00:01:41.750
Now I could just write a 12 here,

00:01:41.750 --> 00:01:43.090
but the half is interesting.

00:01:43.090 --> 00:01:44.840
Because that's going to be my common ratio

00:01:44.840 --> 00:01:46.280
for my geometric series.

00:01:46.280 --> 00:01:48.290
Every successive swing,

00:01:48.290 --> 00:01:51.410
the arc length is half the
arc length of the last swing.

00:01:51.410 --> 00:01:56.140
So it's going to be 24 times 1/2

00:01:56.140 --> 00:01:58.580
and then on the next
swing, it's going to be 24,

00:01:58.580 --> 00:01:59.670
it's going to be half of this.

00:01:59.670 --> 00:02:02.270
So it's going to be 24
times 1/2 times 1/2.

00:02:02.270 --> 00:02:06.400
So that's 24 times 1/2
to the second power.

00:02:06.400 --> 00:02:09.030
And so this would be
the first three swings.

00:02:09.030 --> 00:02:12.350
Notice that the exponent here,
we got to the second power.

00:02:12.350 --> 00:02:14.310
So the first n swings,

00:02:14.310 --> 00:02:16.180
we are going to get to

00:02:16.180 --> 00:02:19.680
24 times 1/2, not to the nth power,

00:02:19.680 --> 00:02:22.170
but to the n minus one power.

00:02:22.170 --> 00:02:23.850
Notice, after two swings, we only get

00:02:23.850 --> 00:02:25.940
to 24 times 1/2 to the first power.

00:02:25.940 --> 00:02:27.580
After three swings, to the second power.

00:02:27.580 --> 00:02:31.250
So after n swings, to
the n minus one power.

00:02:31.250 --> 00:02:33.830
Now, as I said, we don't wanna
just have this expression.

00:02:33.830 --> 00:02:36.670
We actually wanna know,
how do we evaluate this?

00:02:36.670 --> 00:02:38.410
And the way we evaluate this

00:02:38.410 --> 00:02:40.530
is we look at the formula,
which we've explained

00:02:40.530 --> 00:02:42.300
and we've proven in other videos,

00:02:42.300 --> 00:02:45.970
the formula for a finite geometric series.

00:02:45.970 --> 00:02:49.140
So that tells us, and I'll
just write it over here,

00:02:49.140 --> 00:02:52.180
the sum of first

00:02:54.735 --> 00:02:55.818
n terms is a,

00:02:59.025 --> 00:03:00.690
where a is the first term.

00:03:00.690 --> 00:03:03.520
So that's going to be
our 24 in this situation.

00:03:03.520 --> 00:03:06.590
It's a minus a

00:03:06.590 --> 00:03:08.330
times our common ratio,

00:03:08.330 --> 00:03:11.133
I already said that our
common ratio is 1/2,

00:03:12.350 --> 00:03:15.420
to the nth power.

00:03:15.420 --> 00:03:17.420
So one way I like to remember it is,

00:03:17.420 --> 00:03:20.810
it is our first term minus the first term

00:03:20.810 --> 00:03:22.060
that we didn't include,

00:03:22.060 --> 00:03:25.950
or minus what would've been
the term right after this.

00:03:25.950 --> 00:03:29.690
All of that over one
minus our common ratio.

00:03:29.690 --> 00:03:32.410
And there's other ways that
you might've seen this written.

00:03:32.410 --> 00:03:33.890
You could factor an a out,

00:03:33.890 --> 00:03:35.400
and you might have seen
something like this:

00:03:35.400 --> 00:03:38.660
a times one minus r to the n,

00:03:38.660 --> 00:03:42.240
all of that over one minus
r, these two are equivalent.

00:03:42.240 --> 00:03:44.200
But now let's use this.

00:03:44.200 --> 00:03:46.470
So this is going to be equal to,

00:03:46.470 --> 00:03:49.370
actually I'll use this
second form right over here.

00:03:49.370 --> 00:03:51.890
So our first term a is 24.

00:03:51.890 --> 00:03:53.740
So we're going to have 24

00:03:53.740 --> 00:03:57.820
times one minus our common
ratio, which is 1/2,

00:03:57.820 --> 00:04:00.750
to the nth power, well we're talking about

00:04:00.750 --> 00:04:03.390
the first n swing, so I'm
just going to leave an n

00:04:03.390 --> 00:04:04.590
right over there.

00:04:04.590 --> 00:04:09.510
All of that over one
minus our common ratio,

00:04:09.510 --> 00:04:11.970
one minus 1/2.

00:04:11.970 --> 00:04:13.380
So we could leave it like that

00:04:13.380 --> 00:04:16.000
or we could simplify it
a little bit if we like.

00:04:16.000 --> 00:04:19.343
One minus 1/2 is equal to 1/2,

00:04:20.250 --> 00:04:23.210
24 divided by 1/2 is equal to 48,

00:04:23.210 --> 00:04:24.900
so if you wanted to, you could simplify it

00:04:24.900 --> 00:04:26.750
to 48 times one minus 1/2

00:04:28.030 --> 00:04:30.230
to the nth power.

00:04:30.230 --> 00:04:32.710
So either of these would be legitimate.

00:04:32.710 --> 00:04:34.220
Now the second part, they say,

00:04:34.220 --> 00:04:36.390
what is the total distance
the monkey has traveled

00:04:36.390 --> 00:04:40.480
when she completes her 25th swing?

00:04:40.480 --> 00:04:42.800
And they say, round your final
answer to the nearest meter.

00:04:42.800 --> 00:04:46.070
So pause this video and see
if you can work that out.

00:04:46.070 --> 00:04:49.610
All right, well, we can just
use this expression here.

00:04:49.610 --> 00:04:53.520
And we know that we are
completing our 25th swing.

00:04:53.520 --> 00:04:57.390
So n is 25, and so we'll
just put a 25 there.

00:04:57.390 --> 00:05:00.450
So that's going to be
48 times one minus 1/2

00:05:01.480 --> 00:05:04.730
to the 25th power.

00:05:04.730 --> 00:05:07.650
Now this is going to
be a very, very small,

00:05:07.650 --> 00:05:08.770
very, very small number.

00:05:08.770 --> 00:05:10.850
So it's actually going to be pretty close

00:05:10.850 --> 00:05:14.980
to 48 meters, but let's
see what this is equal to.

00:05:14.980 --> 00:05:17.530
And we're going to round
to the nearest meter.

00:05:17.530 --> 00:05:19.450
All right, so let's
get our calculator out.

00:05:19.450 --> 00:05:21.920
And so let's just evaluate 1/2,

00:05:21.920 --> 00:05:23.850
I'll just write that as 0.5

00:05:23.850 --> 00:05:26.870
to the 25th power,

00:05:26.870 --> 00:05:30.000
which, as we said, as we
predicted, is a very small number.

00:05:30.000 --> 00:05:31.590
And then we're going to
subtract that from one,

00:05:31.590 --> 00:05:34.670
so I'll just put a negative
and then I'll add one to it.

00:05:34.670 --> 00:05:37.510
And so that is very close to one.

00:05:37.510 --> 00:05:39.640
And so my prediction is holding true.

00:05:39.640 --> 00:05:42.680
So if I multiply that times 48,

00:05:42.680 --> 00:05:44.150
well, if we round to the nearest meter,

00:05:44.150 --> 00:05:45.950
we get back to 48 meters.

00:05:45.950 --> 00:05:49.400
So this is going to be 48 meters.

00:05:49.400 --> 00:05:50.453
And we're done.