More examples of factoring by grouping

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Kind: captions
Language: en

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In this video, I want to focus
on a few more techniques for

00:00:03.680 --> 00:00:05.210
factoring polynomials.

00:00:05.210 --> 00:00:09.130
And in particular, I want to
focus on quadratics that don't

00:00:09.130 --> 00:00:11.310
have a 1 as the leading
coefficient.

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For example, if I wanted
to factor 4x squared

00:00:15.540 --> 00:00:20.920
plus 25x minus 21.

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Everything we've factored so
far, or all of the quadratics

00:00:23.440 --> 00:00:27.100
we've factored so far, had
either a 1 or negative 1 where

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this 4 is sitting.

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All of a sudden now, we
have this 4 here.

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So what I'm going to teach you
is a technique called,

00:00:32.840 --> 00:00:35.020
factoring by grouping.

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And it's a little bit more
involved than what we've

00:00:37.130 --> 00:00:40.100
learned before, but
it's a neat trick.

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To some degree, it'll become
obsolete once you learn the

00:00:42.210 --> 00:00:44.600
quadratic formula, because,
frankly, the quadratic formula

00:00:44.600 --> 00:00:46.230
is a lot easier.

00:00:46.230 --> 00:00:47.550
But this is how it goes.

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I'll show you the technique.

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And then at the end of this
video, I'll actually show you

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why it works.

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So what we need to do here,
is we need to think of two

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numbers, a and b, where a
times b is equal 4 times

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negative 21.

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So a times b is going to be
equal to 4 times negative 21,

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which is equal to negative 84.

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And those same two numbers,
a plus b, need

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to be equal to 25.

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Let me be very clear.

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This is the 25, so they need
to be equal to 25.

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This is where the 4 is.

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So we go, 4 times negative 21.

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That's a negative 21.

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So what two numbers are there
that would do this?

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Well, we have to look at the
factors of negative 84.

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And once again, one
of these are going

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to have to be positive.

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The other ones are going to have
to be negative, because

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their product is negative.

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So let's think about
the different

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factors that might work.

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4 and negative 21 look
tantalizing, but when you add

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them, you get negative 17.

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Or, if you had negative 4 and
21, you'd get positive 17.

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Doesn't work.

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Let's try some other
combinations.

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1 and 84, too far apart when
you take their difference.

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Because that's essentially what
you're going to do, if

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one is negative and
one is positive.

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Too far apart.

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Let's see you could do 3--
I'm jumping the gun.

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2 and 42.

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Once again, too far apart.

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Negative 2 plus 42 is 40.

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2 plus negative 42 is negative
40-- too far apart.

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3 and-- Let's see, 3 goes into
84-- 3 goes into 8 2 times.

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2 times 3 is 6.

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8 minus 6 is 2.

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Bring down the 4.

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Goes exactly 8 times.

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So 3 and 28.

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This seems interesting.

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And remember, one of these
has to be negative.

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So if we have negative 3 plus
28, that is equal to 25.

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Now, we've found our
two numbers.

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But it's not going to be quite
as simple of an operation as

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what we did when this was
a 1 or negative 1.

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What we're going to do now is
split up this term right here.

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We're going to split it up into
positive 28x minus 3x.

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We're just going to
split that term.

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That term is that term
right there.

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And of course, you have your
minus 21 there, and you have

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your 4x squared over here.

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Now, you might say, how did you
pick the 28 to go here,

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and the negative
3 to go there?

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And it actually does matter.

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The way I thought about it is
3 or negative 3, and 21 or

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negative 21 , they have
some common factors.

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In particular, they have
the factor 3 in common.

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And 28 and 4 have some
common factors.

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So I grouped the 28 on
the side of the 4.

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And you're going to see what
I mean in a second.

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If we, literally, group these
so that term becomes 4x

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squared plus 28x.

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And then, this side, over
here in pink, it's plus

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negative 3x minus 21.

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Once again, I picked these.

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I grouped the negative 3 with
the 21, or the negative 21,

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because they're both
divisible by 3.

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And I grouped the 28 with the
4, because they're both

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divisible by 4.

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And now, in each of these
groups, we factor as

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much out as we can.

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So both of these terms
are divisible by 4x.

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So this orange term is equal
to 4x times x-- 4x squared

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divided by 4x is just x-- plus
28x divided by 4x is just 7.

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Now, this second term.

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Remember, you factor
out everything that

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you can factor out.

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Well, both of these terms are
divisible by 3 or negative 3.

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So let's factor out
a negative 3.

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And this becomes x plus 7.

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And now, something might
pop out at you.

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We have x plus 7 times
4x plus, x plus 7

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times negative 3.

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So we can factor out
an x plus 7.

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This might not be completely
obvious.

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You're probably not
used to factoring

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out an entire binomial.

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But you could view this
could be like a.

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Or if you have 4xa minus 3a,
you would be able to

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factor out an a.

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And I can just leave this
as a minus sign.

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Let me delete this
plus right here.

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Because it's just
minus 3, right?

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Plus negative 3, same
thing as minus 3.

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So what can we do here?

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We have an x plus 7, times 4x.

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We have an x plus 7,
times negative 3.

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Let's factor out the x plus 7.

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We get x plus 7, times
4x minus 3.

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Minus that 3 right there.

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And we've factored
our binomial.

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Sorry, we've factored our
quadratic by grouping.

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And we factored it into
two binomials.

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Let's do another example of
that, because it's a little

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bit involved.

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But once you get the hang
of it's kind of fun.

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So let's say we want to factor
6x squared plus 7x plus 1.

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Same drill.

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We want to find a times b that
is equal to 1 times 6, which

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is equal to 6.

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And we want to find an a plus
b needs to be equal to 7.

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This is a little bit more
straightforward.

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What are the-- well, the obvious
one is 1 and 6, right?

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1 times 6 is 6.

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1 plus 6 is 7.

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So we have a is equal to 1.

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Or let me not even
assign them.

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The numbers here are 1 and 6.

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Now, we want to split this
into a 1x and a 6x.

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But we want to group it so it's
on the side of something

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that it shares a factor with.

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So we're going to have a 6x
squar ed here, plus-- and so

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I'm going to put the
6x first because 6

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and 6 share a factor.

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And then, we're going to
have plus 1x, right?

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6x plus 1x equals 7x .

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That was the whole point.

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They had to add up to 7 .

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And then we have the
final plus 1 there.

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Now, in each of these groups,
we can factor out

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as much as we like.

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So in this first group,
let's factor out a 6x.

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So this first group becomes 6x
times-- 6x squar ed divided by

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6x is just an x.

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6x divided by 6x is just a 1.

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And then, the second
group-- we're going

00:08:06.730 --> 00:08:08.130
to have a plus here.

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But this second group, we just
literally have a x plus 1.

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Or we could even write a
1 times an x plus 1.

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You could imagine I just
factored out of 1 so to speak.

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Now, I have 6x times x plus
1, plus 1 times x plus 1.

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Well, I can factor
out the x plus 1.

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If I factor out an x plus 1,
that's equal to x plus 1 times

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6x plus that 1.

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I'm just doing the

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distributive property in reverse.

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So hopefully you didn't
find that too bad.

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And now, I'm going to actually
explain why this little

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magical system actually works.

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Let me take an example.

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I'll do it in very
general terms.

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Let's say I had ax plus b,
times cx-- actually, I'm

00:09:03.630 --> 00:09:05.300
afraid to use the a's and b's.

00:09:05.300 --> 00:09:07.760
I think that'll confuse
you, because I

00:09:07.760 --> 00:09:08.930
use a's and b's here.

00:09:08.930 --> 00:09:10.240
They won't be the same thing.

00:09:10.240 --> 00:09:13.760
So let me use completely
different letters.

00:09:13.760 --> 00:09:24.000
Let's say I have fx plus g,
times hx plus, I'll use j

00:09:24.000 --> 00:09:25.240
instead of i.

00:09:25.240 --> 00:09:26.450
You'll learn in the future
why don't like

00:09:26.450 --> 00:09:28.710
using i as a variable.

00:09:28.710 --> 00:09:31.000
So what is this going
to be equal to?

00:09:31.000 --> 00:09:35.920
Well, it's going to be fx
times hx which is fhx.

00:09:35.920 --> 00:09:37.710
And then, fx times j.

00:09:37.710 --> 00:09:41.800
So plus fjx.

00:09:41.800 --> 00:09:45.240
And then, we're going
to have g times hx.

00:09:45.240 --> 00:09:49.010
So plus ghx.

00:09:49.010 --> 00:09:50.990
And then g times j.

00:09:50.990 --> 00:09:53.840
Plus gj.

00:09:53.840 --> 00:10:02.500
Or, if we add these two middle
terms, you have fh times x,

00:10:02.500 --> 00:10:08.970
plus-- add these two terms--
fj plus gh x.

00:10:08.970 --> 00:10:12.170
Plus gj.

00:10:12.170 --> 00:10:14.540
Now, what did I do here?

00:10:14.540 --> 00:10:17.550
Well, remember, in all of these
problems where you have

00:10:17.550 --> 00:10:20.650
a non-1 or non-negative 1
coefficient here, we look for

00:10:20.650 --> 00:10:23.560
two numbers that add up to this,
whose product is equal

00:10:23.560 --> 00:10:25.480
to the product of
that times that.

00:10:25.480 --> 00:10:31.780
Well, here we have two numbers
that add up-- let's say that a

00:10:31.780 --> 00:10:33.030
is equal to fj.

00:10:36.970 --> 00:10:37.750
That is a.

00:10:37.750 --> 00:10:40.930
And b is equal to gh.

00:10:40.930 --> 00:10:43.730
So a plus b is going to be
equal to that middle

00:10:43.730 --> 00:10:44.980
coefficient.

00:10:48.370 --> 00:10:52.100
And then what is a times b? a
times b is going to be equal

00:10:52.100 --> 00:10:56.260
to fj times gh.

00:10:59.140 --> 00:11:01.320
We could just reorder these
terms. We're just multiplying

00:11:01.320 --> 00:11:04.430
a bunch of terms. So that could
be rewritten as f times

00:11:04.430 --> 00:11:09.390
h times g times j.

00:11:09.390 --> 00:11:11.090
These are all the same things.

00:11:11.090 --> 00:11:14.620
Well, what is fh times gj?

00:11:14.620 --> 00:11:19.150
This is equal to fh times gj.

00:11:19.150 --> 00:11:22.530
Well, this is equal to the first
coefficient times the

00:11:22.530 --> 00:11:23.670
constant term.

00:11:23.670 --> 00:11:27.870
So a plus b will be equal to
the middle coefficient.

00:11:27.870 --> 00:11:31.930
And a times b will equal the
first coefficient times the

00:11:31.930 --> 00:11:32.670
constant term.

00:11:32.670 --> 00:11:37.330
So that's why this whole
factoring by grouping even

00:11:37.330 --> 00:11:40.180
works, or how we're able
to figure out what

00:11:40.180 --> 00:11:42.200
a and b even are.

00:11:42.200 --> 00:11:44.210
Now, I'm going to close up
with something slightly

00:11:44.210 --> 00:11:45.800
different, but just to make
sure that you have a

00:11:45.800 --> 00:11:49.130
well-rounded education
in factoring things.

00:11:49.130 --> 00:11:52.050
What I want to do is to teach
you to factor things a little

00:11:52.050 --> 00:11:52.940
bit more completely.

00:11:52.940 --> 00:11:55.160
And this is a little
bit of a add-on.

00:11:55.160 --> 00:11:56.660
I was going to make a
whole video on this.

00:11:56.660 --> 00:11:59.250
But I think, on some level,
it might be a

00:11:59.250 --> 00:12:00.480
little obvious for you.

00:12:00.480 --> 00:12:07.190
So let's say we had-- let
me get a good one here.

00:12:07.190 --> 00:12:12.910
Let's say we had negative
x to the third, plus 17x

00:12:12.910 --> 00:12:18.180
squared, minus 70x.

00:12:18.180 --> 00:12:20.130
Immediately, you say, gee, this
isn't even a quadratic.

00:12:20.130 --> 00:12:21.770
I don't know how to solve
something like this.

00:12:21.770 --> 00:12:23.580
It has an x to third power.

00:12:23.580 --> 00:12:26.150
And the first thing you should
realize is that every term

00:12:26.150 --> 00:12:27.840
here is divisible by x.

00:12:27.840 --> 00:12:29.920
So let's factor out an x.

00:12:29.920 --> 00:12:32.260
Or even better, let's factor
out a negative x.

00:12:32.260 --> 00:12:35.320
So if you factor out a negative
x, this is equal to

00:12:35.320 --> 00:12:38.730
negative x times-- negative
x to the third divided by

00:12:38.730 --> 00:12:41.810
negative x is x squared.

00:12:41.810 --> 00:12:47.910
17x squared divided by negative
x is negative 17x.

00:12:47.910 --> 00:12:52.510
Negative 70x divided by negative
x is positive 70.

00:12:52.510 --> 00:12:54.060
The x's cancel out.

00:12:54.060 --> 00:12:56.050
And now, you have something
that might look

00:12:56.050 --> 00:12:59.220
a little bit familiar.

00:12:59.220 --> 00:13:02.530
We have just a standard
quadratic where the leading

00:13:02.530 --> 00:13:03.650
coefficient is a 1.

00:13:03.650 --> 00:13:08.290
We just have to find two numbers
whose product is 70,

00:13:08.290 --> 00:13:11.000
and that add up to
negative 17.

00:13:11.000 --> 00:13:12.920
And the numbers that immediately
jumped into my

00:13:12.920 --> 00:13:17.410
head are negative 10
and negative 7.

00:13:17.410 --> 00:13:19.520
You take their product,
you get 70.

00:13:19.520 --> 00:13:21.870
You add them up, you
get negative 17.

00:13:21.870 --> 00:13:25.920
So this part right here is
going to be x minus 10,

00:13:25.920 --> 00:13:28.410
times x minus 7.

00:13:28.410 --> 00:13:31.540
And of course, you have that
leading negative x.

00:13:31.540 --> 00:13:34.330
The general idea here is just
see if there's anything you

00:13:34.330 --> 00:13:35.320
can factor out.

00:13:35.320 --> 00:13:37.780
And that'll get it into a form
that you might recognize.

00:13:37.780 --> 00:13:38.950
Hopefully, you found
this helpful.

00:13:38.950 --> 00:13:41.145
I want to reiterate what I
showed you at the beginning of

00:13:41.145 --> 00:13:41.510
this video.

00:13:41.510 --> 00:13:44.960
I think it's a really cool
trick, so to speak, to be able

00:13:44.960 --> 00:13:49.470
to factor things that have
a non-1 or non-negative 1

00:13:49.470 --> 00:13:50.460
leading coefficient.

00:13:50.460 --> 00:13:52.720
But to some degree, you're going
to find out easier ways

00:13:52.720 --> 00:13:54.370
to do this, especially
with the quadratic

00:13:54.370 --> 00:13:56.620
formula, in not too long.


More examples of factoring quadratics with a leading coefficient of 1

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Kind: captions
Language: en

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In this video I want to do a
bunch of examples of factoring

00:00:03.870 --> 00:00:06.590
a second degree polynomial,
which is

00:00:06.590 --> 00:00:08.870
often called a quadratic.

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Sometimes a quadratic
polynomial, or just a

00:00:12.650 --> 00:00:15.850
quadratic itself, or quadratic
expression, but all it means

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is a second degree polynomial.

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So something that's going to
have a variable raised to the

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second power.

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In this case, in all of the
examples we'll do, it'll be x.

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So let's say I have the
quadratic expression, x

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squared plus 10x, plus 9.

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And I want to factor it into the
product of two binomials.

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How do we do that?

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Well, let's just think about
what happens if we were to

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take x plus a, and multiply
that by x plus b.

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If we were to multiply these
two things, what happens?

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Well, we have a little bit
of experience doing this.

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This will be x times x, which is
x squared, plus x times b,

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which is bx, plus a times x,
plus a times b-- plus ab.

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Or if we want to add these two
in the middle right here,

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because they're both
coefficients of x.

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We could right this as x squared
plus-- I can write it

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as b plus a, or a plus
b, x, plus ab.

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So in general, if we assume that
this is the product of

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two binomials, we see that this
middle coefficient on the

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x term, or you could say the
first degree coefficient

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there, that's going to be
the sum of our a and b.

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And then the constant term is
going to be the product

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of our a and b.

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Notice, this would map
to this, and this

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would map to this.

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And, of course, this is the
same thing as this.

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So can we somehow pattern
match this to that?

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Is there some a and b where
a plus b is equal to 10?

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And a times b is equal to 9?

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Well, let's just think about
it a little bit.

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What are the factors of 9?

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What are the things that a
and b could be equal to?

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And we're assuming that
everything is an integer.

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And normally when we're
factoring, especially when

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we're beginning to factor,
we're dealing

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with integer numbers.

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So what are the factors of 9?

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They're 1, 3, and 9.

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So this could be a 3 and a 3,
or it could be a 1 and a 9.

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Now, if it's a 3 and a 3, then
you'll have 3 plus 3-- that

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doesn't equal 10.

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But if it's a 1 and a
9, 1 times 9 is 9.

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1 plus 9 is 10.

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So it does work.

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So a could be equal to 1, and
b could be equal to 9.

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So we could factor this
as being x plus 1,

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times x plus 9.

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And if you multiply these two
out, using the skills we

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developed in the last few
videos, you'll see that it is

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indeed x squared plus
10x, plus 9.

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So when you see something like
this, when the coefficient on

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the x squared term, or the
leading coefficient on this

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quadratic is a 1, you can just
say, all right, what two

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numbers add up to this
coefficient right here?

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And those same two numbers, when
you take their product,

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have to be equal to 9.

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And of course, this has to
be in standard form.

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Or if it's not in standard form,
you should put it in

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that form, so that you can
always say, OK, whatever's on

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the first degree coefficient,
my a and b

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have to add to that.

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Whatever's my constant term, my
a times b, the product has

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to be that.

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Let's do several
more examples.

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I think the more examples
we do the more

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sense this'll make.

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Let's say we had x squared
plus 10x, plus-- well, I

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already did 10x, let's do a
different number-- x squared

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plus 15x, plus 50.

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And we want to factor this.

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Well, same drill.

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We have an x squared term.

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We have a first degree term.

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This right here should be
the sum of two numbers.

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And then this term, the constant
term right here,

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should be the product
of two numbers.

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So we need to think of two
numbers that, when I multiply

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them I get 50, and when
I add them, I get 15.

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And this is going to be a bit of
an art that you're going to

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develop, but the more practice
you do, you're going to see

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that it'll start to
come naturally.

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So what could a and b be?

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Let's think about the
factors of 50.

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It could be 1 times 50.

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2 times 25.

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Let's see, 4 doesn't
go into 50.

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It could be 5 times 10.

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I think that's all of them.

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Let's try out these numbers,
and see if any of

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these add up to 15.

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So 1 plus 50 does not
add up to 15.

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2 plus 25 does not
add up to 15.

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But 5 plus 10 does
add up to 15.

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So this could be 5 plus 10, and
this could be 5 times 10.

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So if we were to factor this,
this would be equal to x plus

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5, times x plus 10.

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And multiply it out.

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I encourage you to multiply this
out, and see that this is

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indeed x squared plus
15x, plus 10.

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In fact, let's do it. x
times x, x squared. x

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times 10, plus 10x.

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5 times x, plus 5x.

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5 times 10, plus 50.

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Notice, the 5 times
10 gave us the 50.

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The 5x plus the 10x is giving
us the 15x in between.

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So it's x squared plus
15x, plus 50.

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Let's up the stakes a little
bit, introduce some negative

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signs in here.

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Let's say I had x squared
minus 11x, plus 24.

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Now, it's the exact
same principle.

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I need to think of two numbers,
that when I add them,

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need to be equal
to negative 11.

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a plus b need to be equal
to negative 11.

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And a times b need to
be equal to 24.

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Now, there's something for
you to think about.

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When I multiply both of these
numbers, I'm getting a

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positive number.

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I'm getting a 24.

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That means that both of these
need to be positive, or both

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of these need to be negative.

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That's the only way I'm going to
get a positive number here.

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Now, if when I add them, I get
a negative number, if these

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were positive, there's no way I
can add two positive numbers

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and get a negative number, so
the fact that their sum is

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negative, and the fact that
their product is positive,

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tells me that both a
and b are negative.

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a and b have to be negative.

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Remember, one can't be negative
and the other one

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can't be positive, because the
product would be negative.

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And they both can't be positive,
because when you add

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them it would get you
a positive number.

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So let's just think about
what a and b can be.

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So two negative numbers.

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So let's think about
the factors of 24.

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And we'll kind of have to think
of the negative factors.

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But let me see, it could be 1
times 24, 2 times 11, 3 times

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8, or 4 times 6.

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Now, which of these when I
multiply these-- well,

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obviously when I multiply
1 times 24, I get 24.

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When I get 2 times 11-- sorry,
this is 2 times 12.

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I get 24.

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So we know that all these,
the products are 24.

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But which two of these, which
two factors, when I add them,

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should I get 11?

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And then we could say,
let's take the

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negative of both of those.

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So when you look at these,
3 and 8 jump out.

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3 times 8 is equal to 24.

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3 plus 8 is equal to 11.

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But that doesn't quite
work out, right?

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Because we have a negative
11 here.

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But what if we did negative
3 and negative 8?

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Negative 3 times negative 8
is equal to positive 24.

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Negative 3 plus negative 8
is equal to negative 11.

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So negative 3 and
negative 8 work.

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So if we factor this, x squared
minus 11x, plus 24 is

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going to be equal to x minus
3, times x minus 8.

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Let's do another
one like that.

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Actually, let's mix it
up a little bit.

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Let's say I had x squared
plus 5x, minus 14.

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So here we have a different
situation.

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The product of my two numbers is
negative, right? a times b

00:09:26.460 --> 00:09:28.190
is equal to negative 14.

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My product is negative.

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That tells me that one of them
is positive, and one of them

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is negative.

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And when I add the two, a plus
b, it'd be equal to 5.

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So let's think about
the factors of 14.

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And what combinations of them,
when I add them, if one is

00:09:44.300 --> 00:09:46.560
positive and one is negative,
or I'm really kind of taking

00:09:46.560 --> 00:09:49.780
the difference of the
two, do I get 5?

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So if I take 1 and 14-- I'm just
going to try out things--

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1 and 14, negative 1 plus
14 is negative 13.

00:10:01.820 --> 00:10:04.260
Negative 1 plus 14 is 13.

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So let me write all of the
combinations that I could do.

00:10:07.430 --> 00:10:09.440
And eventually your brain
will just zone in on it.

00:10:09.440 --> 00:10:16.490
So you've got negative 1
plus 14 is equal to 13.

00:10:16.490 --> 00:10:20.460
And 1 plus negative 14 is
equal to negative 13.

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So those don't work.

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That doesn't equal 5.

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Now what about 2 and 7?

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If I do negative 2-- let me do
this in a different color-- if

00:10:29.600 --> 00:10:35.290
I do negative 2 plus 7,
that is equal to 5.

00:10:35.290 --> 00:10:35.750
We're done!

00:10:35.750 --> 00:10:36.670
That worked!

00:10:36.670 --> 00:10:39.440
I mean, we could have tried 2
plus negative 7, but that'd be

00:10:39.440 --> 00:10:41.070
equal to negative 5, so that
wouldn't have worked.

00:10:41.070 --> 00:10:42.960
But negative 2 plus 7 works.

00:10:42.960 --> 00:10:46.590
And negative 2 times
7 is negative 14.

00:10:46.590 --> 00:10:47.600
So there we have it.

00:10:47.600 --> 00:10:53.210
We know it's x minus
2, times x plus 7.

00:10:53.210 --> 00:10:54.330
That's pretty neat.

00:10:54.330 --> 00:10:56.950
Negative 2 times 7
is negative 14.

00:10:56.950 --> 00:11:00.880
Negative 2 plus 7
is positive 5.

00:11:03.760 --> 00:11:07.680
Let's do several more of these,
just to really get well

00:11:07.680 --> 00:11:09.520
honed this skill.

00:11:09.520 --> 00:11:16.360
So let's say we have x squared
minus x, minus 56.

00:11:16.360 --> 00:11:19.570
So the product of the two
numbers have to be minus 56,

00:11:19.570 --> 00:11:21.620
have to be negative 56.

00:11:21.620 --> 00:11:24.430
And their difference, because
one is going to be positive,

00:11:24.430 --> 00:11:26.280
and one is going to be
negative, right?

00:11:26.280 --> 00:11:28.350
Their difference has
to be negative 1.

00:11:28.350 --> 00:11:30.130
And the numbers that immediately
jump out in my

00:11:30.130 --> 00:11:31.900
brain-- and I don't know if they
jump out in your brain,

00:11:31.900 --> 00:11:33.740
we just learned this in
the times tables--

00:11:33.740 --> 00:11:36.480
56 is 8 times 7.

00:11:36.480 --> 00:11:37.480
I mean, there's other numbers.

00:11:37.480 --> 00:11:39.950
It's also 28 times 2.

00:11:39.950 --> 00:11:41.140
It's all sorts of things.

00:11:41.140 --> 00:11:44.300
But 8 times 7 really jumped
out into my brain, because

00:11:44.300 --> 00:11:45.470
they're very close
to each other.

00:11:45.470 --> 00:11:47.730
And we need numbers that are
very close to each other.

00:11:47.730 --> 00:11:50.430
And one of these has to be
positive, and one of these has

00:11:50.430 --> 00:11:51.810
to be negative.

00:11:51.810 --> 00:11:55.250
Now, the fact that when their
sum is negative, tells me that

00:11:55.250 --> 00:11:58.460
the larger of these two should
probably be negative.

00:11:58.460 --> 00:12:01.350
So if we take negative
8 times 7, that's

00:12:01.350 --> 00:12:03.320
equal to negative 56.

00:12:03.320 --> 00:12:08.470
And then if we take negative
8 plus 7, that is equal to

00:12:08.470 --> 00:12:12.100
negative 1, which is exactly the
coefficient right there.

00:12:12.100 --> 00:12:16.490
So when I factor this, this
is going to be x minus 8,

00:12:16.490 --> 00:12:18.690
times x plus 7.

00:12:18.690 --> 00:12:21.695
This is often one of the hardest
concepts people learn

00:12:21.695 --> 00:12:23.920
in algebra, because it
is a bit of an art.

00:12:23.920 --> 00:12:26.710
You have to look at all of the
factors here, play with the

00:12:26.710 --> 00:12:29.710
positive and negative signs,
see which of those factors

00:12:29.710 --> 00:12:31.900
when one is positive, one is
negative, add up to the

00:12:31.900 --> 00:12:33.590
coefficient on the x term.

00:12:33.590 --> 00:12:35.860
But as you do more and more
practice, you'll see that

00:12:35.860 --> 00:12:39.280
it'll become a bit
of second nature.

00:12:39.280 --> 00:12:42.350
Now let's step up the stakes
a little bit more.

00:12:42.350 --> 00:12:46.140
Let's say we had negative x
squared-- everything we've

00:12:46.140 --> 00:12:49.040
done so far had a positive
coefficient, a positive 1

00:12:49.040 --> 00:12:50.690
coefficient on the
x squared term.

00:12:50.690 --> 00:12:55.590
But let's say we had a
negative x squared

00:12:55.590 --> 00:12:59.440
minus 5x, plus 24.

00:12:59.440 --> 00:13:00.910
How do we do this?

00:13:00.910 --> 00:13:03.420
Well, the easiest way I can
think of doing it is factor

00:13:03.420 --> 00:13:05.670
out a negative 1, and then it
becomes just like the problems

00:13:05.670 --> 00:13:07.260
we've been doing before.

00:13:07.260 --> 00:13:11.660
So this is the same thing as
negative 1 times positive x

00:13:11.660 --> 00:13:15.990
squared, plus 5x, minus 24.

00:13:15.990 --> 00:13:16.300
Right?

00:13:16.300 --> 00:13:18.080
I just factored a
negative 1 out.

00:13:18.080 --> 00:13:20.180
You can multiply negative 1
times all of these, and you'll

00:13:20.180 --> 00:13:21.690
see it becomes this.

00:13:21.690 --> 00:13:23.555
Or you could factor the negative
1 out and divide all

00:13:23.555 --> 00:13:25.000
of these by negative 1.

00:13:25.000 --> 00:13:26.760
And you get that right there.

00:13:26.760 --> 00:13:29.360
Now, same game as before.

00:13:29.360 --> 00:13:33.510
I need two numbers, that when
I take their product I get

00:13:33.510 --> 00:13:34.690
negative 24.

00:13:34.690 --> 00:13:37.125
So one will be positive,
one will be negative.

00:13:41.550 --> 00:13:43.770
When I take their sum,
it's going to be 5.

00:13:43.770 --> 00:13:48.850
So let's think about
24 is 1 and 24.

00:13:48.850 --> 00:13:55.750
Let's see, if this is negative 1
and 24, it'd be positive 23,

00:13:55.750 --> 00:13:57.540
if it was the other way around,
it'd be negative 23.

00:13:57.540 --> 00:13:58.500
Doesn't work.

00:13:58.500 --> 00:14:01.210
What about 2 and 12?

00:14:01.210 --> 00:14:04.530
Well, if this is negative--
remember, one of these has to

00:14:04.530 --> 00:14:05.040
be negative.

00:14:05.040 --> 00:14:07.570
If the 2 is negative, their
sum would be 10.

00:14:07.570 --> 00:14:09.750
If the 12 is negative, their
sum would be negative 10.

00:14:09.750 --> 00:14:11.300
Still doesn't work.

00:14:11.300 --> 00:14:13.290
3 and 8.

00:14:13.290 --> 00:14:16.850
If the 3 is negative,
their sum will be 5.

00:14:16.850 --> 00:14:17.910
So it works!

00:14:17.910 --> 00:14:24.620
So if we pick negative 3 and
8, negative 3 and 8 work.

00:14:24.620 --> 00:14:26.630
Because negative
3 plus 8 is 5.

00:14:26.630 --> 00:14:29.510
Negative 3 times 8
is negative 24.

00:14:29.510 --> 00:14:31.640
So this is going to be equal
to-- can't forget that

00:14:31.640 --> 00:14:35.230
negative 1 out front, and then
we factor the inside.

00:14:35.230 --> 00:14:42.990
Negative 1 times x minus
3, times x plus 8.

00:14:42.990 --> 00:14:44.800
And if you really wanted to,
you could multiply the

00:14:44.800 --> 00:14:46.450
negative 1 times this,
you would get 3

00:14:46.450 --> 00:14:47.560
minus x if you did.

00:14:47.560 --> 00:14:48.810
Or you don't have to.

00:14:51.810 --> 00:14:53.220
Let's do one more of these.

00:14:53.220 --> 00:14:56.060
The more practice, the
better, I think.

00:14:56.060 --> 00:15:01.630
All right, let's say I had
negative x squared

00:15:01.630 --> 00:15:06.690
plus 18x, minus 72.

00:15:06.690 --> 00:15:09.320
So once again, I like to factor
out the negative 1.

00:15:09.320 --> 00:15:13.040
So this is equal to negative
1 times x squared,

00:15:13.040 --> 00:15:16.910
minus 18x, plus 72.

00:15:16.910 --> 00:15:19.650
Now we just have to think of
two numbers, that when I

00:15:19.650 --> 00:15:22.330
multiply them I get
positive 72.

00:15:22.330 --> 00:15:24.480
So they have to be
the same sign.

00:15:24.480 --> 00:15:26.930
And that makes it easier in our
head, at least in my head.

00:15:26.930 --> 00:15:29.260
When I multiply them,
I get positive 72.

00:15:29.260 --> 00:15:32.040
When I add them, I
get negative 18.

00:15:32.040 --> 00:15:34.410
So they're the same sign, and
their sum is a negative

00:15:34.410 --> 00:15:36.470
number, they both must
be negative.

00:15:41.470 --> 00:15:43.790
And we could go through all
of the factors of 72.

00:15:43.790 --> 00:15:48.510
But the one that springs up,
maybe you think of 8 times 9,

00:15:48.510 --> 00:15:53.270
but 8 times 9, or negative 8
minus 9, or negative 8 plus

00:15:53.270 --> 00:15:55.210
negative 9, doesn't work.

00:15:55.210 --> 00:15:58.060
That turns into 17.

00:15:58.060 --> 00:15:58.860
That was close.

00:15:58.860 --> 00:15:59.560
Let me show you that.

00:15:59.560 --> 00:16:04.400
Negative 9 plus negative 8, that
is equal to negative 17.

00:16:04.400 --> 00:16:06.140
Close, but no cigar.

00:16:06.140 --> 00:16:06.990
So what other ones are there?

00:16:06.990 --> 00:16:08.005
We have 6 and 12.

00:16:08.005 --> 00:16:09.705
That actually seems
pretty good.

00:16:09.705 --> 00:16:13.710
If we have negative 6 plus
negative 12, that is equal to

00:16:13.710 --> 00:16:15.360
negative 18.

00:16:15.360 --> 00:16:16.590
Notice, it's a bit of an art.

00:16:16.590 --> 00:16:18.810
You have to try the different
factors here.

00:16:18.810 --> 00:16:22.360
So this will become negative
1-- don't want to forget

00:16:22.360 --> 00:16:29.440
that-- times x minus 6,
times x minus 12.


Factoring special products

WEBVTT
Kind: captions
Language: en

00:00:00.420 --> 00:00:03.860
Let's solve some quadratic
equations by factoring.

00:00:03.860 --> 00:00:12.490
So let's say I had x squared
plus 4x is equal to 21.

00:00:12.490 --> 00:00:16.260
Now your impulse might be to
try to factor out an x and

00:00:16.260 --> 00:00:18.250
somehow set that equal to 21.

00:00:18.250 --> 00:00:21.650
And that will not lead you
to good solutions.

00:00:21.650 --> 00:00:22.670
You'll probably end up doing

00:00:22.670 --> 00:00:24.880
something that's not justified.

00:00:24.880 --> 00:00:27.910
What you need to do here is
put the entire quadratic

00:00:27.910 --> 00:00:30.460
expression on one side
of the equation.

00:00:30.460 --> 00:00:32.460
We'll do it onto the
left-hand side.

00:00:32.460 --> 00:00:37.710
So let's subtract 21 from both
sides of this equation.

00:00:37.710 --> 00:00:40.680
The left-hand side then
becomes x squared

00:00:40.680 --> 00:00:44.080
plus 4x minus 21.

00:00:44.080 --> 00:00:47.130
And then the right-hand side
will be equal to 0.

00:00:47.130 --> 00:00:48.710
And the way you want
to solve this, this

00:00:48.710 --> 00:00:50.170
is a quadratic equation.

00:00:50.170 --> 00:00:54.230
We have a quadratic expression
being set equal to 0.

00:00:54.230 --> 00:00:56.560
The way you want to solve this
is you want to factor them,

00:00:56.560 --> 00:00:58.500
and say, OK, each of
those factors could

00:00:58.500 --> 00:01:00.470
then be equal to 0.

00:01:00.470 --> 00:01:02.040
So how do we factor this?

00:01:02.040 --> 00:01:04.910
Well, we saw in the last video
that we have to figure out two

00:01:04.910 --> 00:01:10.040
numbers whose product is equal
to negative 21, and whose sum

00:01:10.040 --> 00:01:11.580
is equal to 4.

00:01:11.580 --> 00:01:15.900
This would be a plus b would
have to be equal to 4.

00:01:15.900 --> 00:01:17.990
Since their product is negative,
they have to be of

00:01:17.990 --> 00:01:19.270
different signs.

00:01:19.270 --> 00:01:21.050
And so let's see, the number
that jumps out

00:01:21.050 --> 00:01:23.530
at me is 7 and 3.

00:01:23.530 --> 00:01:26.280
If I have negative 7 and
positive 3, I would get

00:01:26.280 --> 00:01:27.580
negative 4.

00:01:27.580 --> 00:01:30.210
So let's do positive
7 and negative 3.

00:01:30.210 --> 00:01:33.180
So the a and b are positive
7 and negative 3.

00:01:33.180 --> 00:01:36.250
When I take the product,
I get negative 21.

00:01:36.250 --> 00:01:38.660
When I take their sum,
I get positive 4.

00:01:38.660 --> 00:01:41.070
So I can rewrite this
equation here.

00:01:41.070 --> 00:01:47.180
I could rewrite it as x
plus 7, times x minus

00:01:47.180 --> 00:01:51.150
3, is equal to 0.

00:01:51.150 --> 00:01:53.680
And now I can solve this by
saying, look, I have two

00:01:53.680 --> 00:01:54.690
quantities.

00:01:54.690 --> 00:01:56.040
Their product is equal to 0.

00:01:56.040 --> 00:01:59.610
That means that one or both of
them have to be equal to 0.

00:01:59.610 --> 00:02:02.770
So that means that x plus
7 is equal to 0.

00:02:02.770 --> 00:02:04.040
That's an x.

00:02:04.040 --> 00:02:07.580
Or x minus 3 is equal to 0.

00:02:07.580 --> 00:02:10.419
I could subtract 7 from both
sides of this equation.

00:02:10.419 --> 00:02:12.990
And I would get x is equal
to negative 7.

00:02:12.990 --> 00:02:16.270
And over here, I can add 3 to
both sides of this equation.

00:02:16.270 --> 00:02:19.320
And I'll get x is equal to 3.

00:02:19.320 --> 00:02:23.540
So both of these numbers are
solutions to this equation.

00:02:23.540 --> 00:02:24.750
You could try it out.

00:02:24.750 --> 00:02:30.300
If you do 7-- negative
7 squared is 49.

00:02:30.300 --> 00:02:34.200
Negative 7 times 4 is minus
28, or negative 28.

00:02:34.200 --> 00:02:37.010
And that does indeed equal 21.

00:02:37.010 --> 00:02:40.280
And I'll let you try it out
with the positive 3.

00:02:40.280 --> 00:02:41.290
Actually, let's just do it.

00:02:41.290 --> 00:02:44.190
3 squared is 9, plus
4 times 3 is 12.

00:02:44.190 --> 00:02:47.170
9 plus 12 is, indeed, 21.

00:02:47.170 --> 00:02:49.920
Let's do a bunch
more examples.

00:02:49.920 --> 00:02:58.300
Let's say I have x squared
plus 49 is equal to 14x.

00:02:58.300 --> 00:03:01.090
Once again, whenever you see
anything like this, get all of

00:03:01.090 --> 00:03:04.140
your terms on one side of the
equation and get a 0 on the

00:03:04.140 --> 00:03:04.540
other side.

00:03:04.540 --> 00:03:08.410
That's the best way to solve
a quadratic equation.

00:03:08.410 --> 00:03:11.090
So let's subtract 14x
from both sides.

00:03:11.090 --> 00:03:17.430
We could write this as x squared
minus 14x plus 49 is

00:03:17.430 --> 00:03:18.570
equal to 0.

00:03:18.570 --> 00:03:21.140
Obviously, 14x minus 14x is 0.

00:03:21.140 --> 00:03:24.780
This quantity minus 14x is this
quantity right there.

00:03:24.780 --> 00:03:28.590
Now we just have to think about
what two numbers, when I

00:03:28.590 --> 00:03:32.580
take their product, I'm going
to get 49, and when I take

00:03:32.580 --> 00:03:36.670
their sum, I'm going
to get negative 14.

00:03:36.670 --> 00:03:39.520
So one, they have to be the same
sign because this is a

00:03:39.520 --> 00:03:41.880
positive number right here.

00:03:41.880 --> 00:03:43.290
And they're both going to
be negative because

00:03:43.290 --> 00:03:44.940
their sum is negative.

00:03:44.940 --> 00:03:46.480
And there's something
interesting here.

00:03:46.480 --> 00:03:48.400
49 is a perfect square.

00:03:48.400 --> 00:03:53.890
Its factors are 1, 7, and 49.

00:03:53.890 --> 00:03:56.020
So maybe 7 will work, or
even better, maybe

00:03:56.020 --> 00:03:57.530
negative 7 will work.

00:03:57.530 --> 00:03:58.410
And it does!

00:03:58.410 --> 00:04:03.980
Negative 7 times negative
7 is 49.

00:04:03.980 --> 00:04:08.100
And negative 7 plus negative
7 is negative 14.

00:04:08.100 --> 00:04:10.540
We have that pattern there,
where we have 2 times a

00:04:10.540 --> 00:04:12.250
number, and then we have
the number squared.

00:04:12.250 --> 00:04:14.000
This is a perfect square.

00:04:14.000 --> 00:04:19.209
This is equal to x minus
7, times x minus

00:04:19.209 --> 00:04:21.839
7, is equal to 0.

00:04:21.839 --> 00:04:22.750
Don't want to forget that.

00:04:22.750 --> 00:04:28.000
Or we could write this as x
minus 7 squared is equal to 0.

00:04:28.000 --> 00:04:33.120
So this was a perfect square
of a binomial.

00:04:33.120 --> 00:04:36.010
And if x minus 7 squared is
equal to 0, take the square

00:04:36.010 --> 00:04:37.220
root of both sides.

00:04:37.220 --> 00:04:39.990
You'll get x minus
7 is equal to 0.

00:04:39.990 --> 00:04:43.890
I mean, you could say x minus
7 is 0 or x minus 7 is 0.

00:04:43.890 --> 00:04:45.210
But that'd be redundant.

00:04:45.210 --> 00:04:47.640
So we just get x minus 7 is 0.

00:04:47.640 --> 00:04:52.400
Add 7 to both sides, and you
get x is equal to 7.

00:04:52.400 --> 00:04:56.620
Only one solution there.

00:04:56.620 --> 00:05:02.240
Let's do another one in pink.

00:05:02.240 --> 00:05:09.350
Let's say we have x squared
minus 64 is equal to 0.

00:05:09.350 --> 00:05:11.630
Now this looks interesting
right here.

00:05:15.160 --> 00:05:16.780
A bell might be ringing
in your head on

00:05:16.780 --> 00:05:18.010
how to solve this.

00:05:18.010 --> 00:05:20.300
This has no x term, but
we could think of it

00:05:20.300 --> 00:05:21.410
as having an x term.

00:05:21.410 --> 00:05:27.660
We could rewrite this as x
squared plus 0x minus 64.

00:05:27.660 --> 00:05:30.070
So in this situation, we could
say, OK, what two numbers,

00:05:30.070 --> 00:05:33.130
when I multiply them,
equal 64, and when I

00:05:33.130 --> 00:05:35.580
add them equal 0?

00:05:35.580 --> 00:05:37.720
And when I take their product,
I'm getting a

00:05:37.720 --> 00:05:38.980
negative number, right?

00:05:38.980 --> 00:05:40.080
This is a times b.

00:05:40.080 --> 00:05:41.220
It's a negative number.

00:05:41.220 --> 00:05:43.195
So that must mean that they
have opposite signs.

00:05:46.600 --> 00:05:48.590
And when I add them, I get 0.

00:05:48.590 --> 00:05:53.740
That must mean that a plus minus
b is equal to 0, or that

00:05:53.740 --> 00:05:57.870
a is equal to b, that we're
dealing with the same number.

00:05:57.870 --> 00:06:00.500
We're essentially dealing with
the same number, the negatives

00:06:00.500 --> 00:06:01.790
of each other.

00:06:01.790 --> 00:06:03.270
So what can it be?

00:06:03.270 --> 00:06:05.650
Well, if we're doing the same
number and they're negatives

00:06:05.650 --> 00:06:14.505
of each other, 64 is
exactly 8 squared.

00:06:14.505 --> 00:06:17.300
But it's negative 64, so maybe
we're dealing with one

00:06:17.300 --> 00:06:20.010
negative 8, and we're dealing
with one positive 8.

00:06:20.010 --> 00:06:22.250
And if we add those two
together, we do

00:06:22.250 --> 00:06:23.720
indeed get to 0.

00:06:23.720 --> 00:06:29.840
So this will be x minus
8 times x plus 8.

00:06:29.840 --> 00:06:31.300
Now you don't always have
to go through this

00:06:31.300 --> 00:06:32.250
process I did here.

00:06:32.250 --> 00:06:36.790
You might already remember that
if I have a plus b times

00:06:36.790 --> 00:06:42.000
a minus b, that that's equal to
a squared minus b squared.

00:06:42.000 --> 00:06:45.400
So if you see something that
fits the pattern, a squared

00:06:45.400 --> 00:06:49.100
minus b squared, you could
immediately say, oh, that's

00:06:49.100 --> 00:06:58.330
going to be a plus b-- a is x,
b is 8-- times a minus b.

00:06:58.330 --> 00:07:02.720
Let's do a couple more of just
general problems. I won't tell

00:07:02.720 --> 00:07:06.270
you what type these
are going to be.

00:07:06.270 --> 00:07:07.110
Let me switch colors.

00:07:07.110 --> 00:07:08.490
It's getting monotonous.

00:07:08.490 --> 00:07:14.280
Let's say we have x squared
minus 24x plus

00:07:14.280 --> 00:07:17.530
144 is equal to 0.

00:07:17.530 --> 00:07:20.650
Well, 144 is conspicuously
12 squared.

00:07:20.650 --> 00:07:24.290
And this is conspicuously
2 times negative 12.

00:07:24.290 --> 00:07:26.570
Or this is conspicuously
negative 12 squared.

00:07:26.570 --> 00:07:29.510
So this is negative 12
times negative 12.

00:07:29.510 --> 00:07:32.400
This is negative 12
plus negative 12.

00:07:32.400 --> 00:07:38.060
So this expression can be
rewritten as x minus 12 times

00:07:38.060 --> 00:07:43.050
x minus 12, or x minus
12 squared.

00:07:43.050 --> 00:07:45.550
We're going to set
that equal to 0.

00:07:45.550 --> 00:07:48.790
This is going to be 0 when
x minus 12 is equal to 0.

00:07:48.790 --> 00:07:52.230
You can say either of these
could be equal to 0, but

00:07:52.230 --> 00:07:53.510
they're the same thing.

00:07:53.510 --> 00:07:57.580
Add 12 to both sides of that
equation and you get x is

00:07:57.580 --> 00:07:58.560
equal to 12.

00:07:58.560 --> 00:08:02.020
And I just realized, this
problem up here, I factored

00:08:02.020 --> 00:08:04.830
it, but I didn't actually
solve the equation.

00:08:04.830 --> 00:08:07.200
So this has to be equal to 0.

00:08:07.200 --> 00:08:10.070
Let's take a step back to
this equation up here.

00:08:10.070 --> 00:08:13.470
And the only way that this thing
over here will be 0 is

00:08:13.470 --> 00:08:18.670
if either x minus 8 is
equal to 0 or x plus

00:08:18.670 --> 00:08:21.110
8 is equal to 0.

00:08:21.110 --> 00:08:23.200
So add 8 to both
sides of this.

00:08:23.200 --> 00:08:25.100
You get x could be equal to 8.

00:08:25.100 --> 00:08:27.460
Subtract 8 from both
sides of this.

00:08:27.460 --> 00:08:32.770
You get x could also be
equal to negative 8.

00:08:32.770 --> 00:08:33.820
Let's do one more.

00:08:33.820 --> 00:08:38.159
Just to really, really get the
point drilled in your head.

00:08:38.159 --> 00:08:39.650
Let's do one more.

00:08:39.650 --> 00:08:47.180
Let's say we have 4x squared
minus 25 is equal to 0.

00:08:47.180 --> 00:08:49.870
So you might already
see the pattern.

00:08:49.870 --> 00:08:52.870
This is an a squared.

00:08:52.870 --> 00:08:54.780
This is a b squared.

00:08:54.780 --> 00:08:58.180
We have the pattern of a squared
minus b squared,

00:08:58.180 --> 00:09:02.140
where, in this case, a would
be equal to x, right?

00:09:02.140 --> 00:09:04.020
This is 2x squared.

00:09:04.020 --> 00:09:07.230
And b would be equal to 5.

00:09:07.230 --> 00:09:12.570
So if you have a squared minus b
squared, this is going to be

00:09:12.570 --> 00:09:16.820
equal to a plus b
times a minus b.

00:09:16.820 --> 00:09:20.590
In this situation, that means
that 4x squared minus 25 is

00:09:20.590 --> 00:09:26.660
going to be 2x plus 5
times 2x minus 5.

00:09:26.660 --> 00:09:29.520
And of course, that will
be equal to 0.

00:09:29.520 --> 00:09:34.400
And this will only be equal to 0
if either 2x plus 5 is equal

00:09:34.400 --> 00:09:39.870
to 0 or 2x minus 5
is equal to 0.

00:09:39.870 --> 00:09:42.500
And then we can solve
each of these.

00:09:42.500 --> 00:09:44.210
Subtract 5 from both sides.

00:09:44.210 --> 00:09:46.550
You get 2x is equal
to negative 5.

00:09:46.550 --> 00:09:48.280
Divide both sides by 2.

00:09:48.280 --> 00:09:52.030
You could get one solution
is negative 5/2.

00:09:52.030 --> 00:09:53.950
Over here, add 5
to both sides.

00:09:53.950 --> 00:09:58.020
You get 2x is equal
to positive 5.

00:09:58.020 --> 00:09:59.420
Divide both sides by 2.

00:09:59.420 --> 00:10:03.310
You get x could also be
equal to positive 5/2.

00:10:03.310 --> 00:10:07.920
So both of these satisfy
that equation up there.


Polynomial Equations in Factored Form

WEBVTT
Kind: captions
Language: en

00:00:00.480 --> 00:00:03.080
Everything we've done with
polynomials so far has been

00:00:03.080 --> 00:00:05.410
either adding or multiplying
them.

00:00:05.410 --> 00:00:06.920
But what if we wanted
to go the other way?

00:00:06.920 --> 00:00:09.960
What if we actually wanted
to factor a polynomial?

00:00:09.960 --> 00:00:11.430
Let me show you what I mean.

00:00:11.430 --> 00:00:14.670
So let's have the polynomial
3x to the

00:00:14.670 --> 00:00:18.470
third power minus 21x.

00:00:18.470 --> 00:00:20.770
When I say factor it, I
essentially want to break it

00:00:20.770 --> 00:00:24.670
up into the product of, maybe
polynomials, it could be a

00:00:24.670 --> 00:00:27.290
mononomial, which is
just one term.

00:00:27.290 --> 00:00:31.980
And to do that over here, we
could factor out a 3x here.

00:00:31.980 --> 00:00:38.230
So if you factor out a 3x this
is equal to 3x times-- what's

00:00:38.230 --> 00:00:42.460
3x to the third divided by 3x?

00:00:42.460 --> 00:00:43.490
Let me write this
way actually.

00:00:43.490 --> 00:00:53.030
This is the same thing as 3x
times 1/3 x times 3x to the

00:00:53.030 --> 00:00:56.160
third minus 21x.

00:00:56.160 --> 00:00:59.440
Notice, 3x times 1/3x, I'm
not changing the number.

00:00:59.440 --> 00:01:01.830
This is 3x/3x, this is 1.

00:01:01.830 --> 00:01:05.190
But the reason why I picked 3x
is because I saw that both of

00:01:05.190 --> 00:01:07.070
these terms are divisible
by 3x.

00:01:07.070 --> 00:01:09.340
And you're going to see
that when I divide

00:01:09.340 --> 00:01:11.340
both of them by 3x.

00:01:11.340 --> 00:01:16.280
So this is going to be the same
thing as 3x times 3x to

00:01:16.280 --> 00:01:22.700
the third over 3x
minus 21x/3x.

00:01:22.700 --> 00:01:25.160
And now this is going
to be equal to 3x.

00:01:25.160 --> 00:01:26.570
What's this first
term equal to?

00:01:26.570 --> 00:01:28.200
The 3's cancel out.

00:01:28.200 --> 00:01:30.910
x to the third divided by x.

00:01:30.910 --> 00:01:33.175
We learned in our exponent
rules, you could just subtract

00:01:33.175 --> 00:01:34.200
the exponents there.

00:01:34.200 --> 00:01:37.540
That's x times x times x divided
by x, just going to be

00:01:37.540 --> 00:01:39.880
x times x, or just x squared.

00:01:39.880 --> 00:01:44.390
So it's going to be, this first
term here is just going

00:01:44.390 --> 00:01:46.580
to be an x squared.

00:01:46.580 --> 00:01:48.910
And then the x's cancel
out here.

00:01:48.910 --> 00:01:50.630
21 divided by 3 is 7.

00:01:50.630 --> 00:01:53.390
x squared minus 7.

00:01:53.390 --> 00:01:55.360
And if you multiply them out,
you're going to see you're

00:01:55.360 --> 00:01:57.810
going to get 3x to the
third minus 21x.

00:01:57.810 --> 00:01:59.740
Now I went through all of these
steps really, just to

00:01:59.740 --> 00:02:01.400
show you exactly
what I'm doing.

00:02:01.400 --> 00:02:04.810
But as we progress, you're going
to see that your brain

00:02:04.810 --> 00:02:06.710
will just automatically say,
hey, both of these are

00:02:06.710 --> 00:02:10.440
divisible by 3x, let me factor
out a 3x, and then divide both

00:02:10.440 --> 00:02:14.320
of these terms by 3x to
get x squared minus 7.

00:02:14.320 --> 00:02:15.920
Let's do another one.

00:02:15.920 --> 00:02:21.830
Let's say we had 4x to
the third plus 10x

00:02:21.830 --> 00:02:24.950
squared minus 2x.

00:02:24.950 --> 00:02:27.500
So when I look at this,
immediately what pops out is

00:02:27.500 --> 00:02:31.040
that all of these terms are
divisible by 2, and all of

00:02:31.040 --> 00:02:33.170
them are divisible by x.

00:02:33.170 --> 00:02:36.130
So let me factor out a 2x.

00:02:36.130 --> 00:02:41.110
So this is equal to 2x times--
and when we factor out a 2x

00:02:41.110 --> 00:02:43.480
let's divide each of
these terms by 2x.

00:02:43.480 --> 00:02:45.990
4x third divided by 2x.

00:02:45.990 --> 00:02:49.010
4 divided by 2 is 2.

00:02:49.010 --> 00:02:53.390
And then x third divided
by x is x squared.

00:02:53.390 --> 00:02:58.070
And then we have plus 10x
squared divided by 2x.

00:02:58.070 --> 00:03:01.190
10 divided by 2 is 5.

00:03:01.190 --> 00:03:04.850
x squared divided by x is x.

00:03:04.850 --> 00:03:08.490
And then finally, negative
2x divided by 2x is

00:03:08.490 --> 00:03:10.170
just negative 1.

00:03:10.170 --> 00:03:12.080
And you know, if you don't
believe me, just

00:03:12.080 --> 00:03:12.980
multiply this out.

00:03:12.980 --> 00:03:15.480
Distribute the 2x and you
will get exactly this.

00:03:15.480 --> 00:03:18.510
So all we see, we try to look
for something that's divisible

00:03:18.510 --> 00:03:19.590
into every term.

00:03:19.590 --> 00:03:22.940
We factor it out, factoring it
out, we put it outside of a

00:03:22.940 --> 00:03:24.610
parentheses, so to speak.

00:03:24.610 --> 00:03:28.250
And then we divide each of those
terms by that factor.

00:03:28.250 --> 00:03:31.020
We did that explicitly
up here.

00:03:31.020 --> 00:03:34.020
Now you're probably saying,
hey Sal, why are we even

00:03:34.020 --> 00:03:36.390
worried about doing this?

00:03:36.390 --> 00:03:41.170
And the answer is, is it really
helps us solve a lot of

00:03:41.170 --> 00:03:44.790
certain types of polynomial
equations.

00:03:44.790 --> 00:03:46.760
Let me show you what I mean.

00:03:46.760 --> 00:03:52.350
Let's say we have the polynomial
equation 9x squared

00:03:52.350 --> 00:03:55.140
is equal to 27x.

00:03:55.140 --> 00:03:57.630
And we want to make sure that
we get all of the x's that

00:03:57.630 --> 00:03:59.400
satisfy this equation.

00:03:59.400 --> 00:04:02.485
So let me subtract 27x
from both sides.

00:04:07.080 --> 00:04:09.280
So I get something
that equals 0.

00:04:09.280 --> 00:04:11.600
And this will be a general trend
whenever you're doing

00:04:11.600 --> 00:04:15.510
polynomial equations, very
oftentimes you will.

00:04:15.510 --> 00:04:20.430
The left-hand side becomes
9x squared minus 27x.

00:04:20.430 --> 00:04:23.380
The right-hand side becomes 0.

00:04:23.380 --> 00:04:26.930
And now over here, we can see
that both of these terms are

00:04:26.930 --> 00:04:29.090
divisible by 9x, right?

00:04:29.090 --> 00:04:32.100
27 is divisible by 9, so is 9.

00:04:32.100 --> 00:04:34.370
And they're both
divisible by x.

00:04:34.370 --> 00:04:39.140
So let's factor out a 9x.

00:04:39.140 --> 00:04:45.140
So 9x squared divided
by 9x is just an x.

00:04:45.140 --> 00:04:48.920
And then 27x divided by 9x.

00:04:48.920 --> 00:04:52.760
That's negative 27/9
is negative 3.

00:04:52.760 --> 00:04:55.530
And then the x divided
by x is just a 1.

00:04:55.530 --> 00:04:59.390
So this is 9x times x minus 3--
you can multiply it out to

00:04:59.390 --> 00:05:01.880
verify that these are the
same thing-- is going

00:05:01.880 --> 00:05:04.310
to be equal to 0.

00:05:04.310 --> 00:05:06.680
Now this part is interesting.

00:05:06.680 --> 00:05:09.750
When I take the product of two
numbers, and I say they're

00:05:09.750 --> 00:05:11.810
equal 0-- let me do a
little aside here.

00:05:11.810 --> 00:05:15.400
If I were tell you that a times
b is equal to 0, what

00:05:15.400 --> 00:05:16.630
does that tell us?

00:05:16.630 --> 00:05:26.140
That tells us that either
a or b or both are 0.

00:05:26.140 --> 00:05:28.190
At least one of these
two have to be 0.

00:05:28.190 --> 00:05:30.750
Maybe both of them are going
to be equal to 0.

00:05:30.750 --> 00:05:32.340
So we're doing the exact
same thing here.

00:05:32.340 --> 00:05:34.940
We're taking the product-- you
can view that as a, you could

00:05:34.940 --> 00:05:37.580
view that as b-- we're taking
the product of two numbers and

00:05:37.580 --> 00:05:40.330
saying that that product
is equal to 0.

00:05:40.330 --> 00:05:44.790
So that tells us that either 9x
is going to be equal to 0,

00:05:44.790 --> 00:05:50.030
or x minus 3 is going
to be equal to 0.

00:05:50.030 --> 00:05:53.320
Either of these, or both of
these, are going to equal 0.

00:05:53.320 --> 00:05:54.600
This we know how to solve.

00:05:54.600 --> 00:05:56.780
Divide both sides by 9.

00:05:56.780 --> 00:05:59.260
You get x is equal to 0.

00:05:59.260 --> 00:06:00.510
This we know how to solve.

00:06:00.510 --> 00:06:01.760
Add 3 to both sides.

00:06:04.410 --> 00:06:07.660
And you're left with x-- because
these cancel out-- is

00:06:07.660 --> 00:06:09.080
equal to 3.

00:06:09.080 --> 00:06:12.390
So this equation here actually
had two solutions. x could be

00:06:12.390 --> 00:06:13.930
0, and you can verify that.

00:06:13.930 --> 00:06:15.730
0 is equal to 0.

00:06:15.730 --> 00:06:17.110
Or x could be equal to 3.

00:06:17.110 --> 00:06:22.160
9 times 9 is the same
thing as 27 times 3.

00:06:22.160 --> 00:06:23.570
And you might have been able
to figure that out in your

00:06:23.570 --> 00:06:26.280
head, but when you do it this
way you make sure that you get

00:06:26.280 --> 00:06:28.460
all of the solutions
to the problem.

00:06:28.460 --> 00:06:30.420
Let's do one more like this.

00:06:30.420 --> 00:06:38.440
Let's say I had 4a squared
plus a is equal to 0.

00:06:38.440 --> 00:06:40.500
Once again, we can
factor an a out.

00:06:40.500 --> 00:06:42.590
Both of these terms are
divisible by a, so this is the

00:06:42.590 --> 00:06:47.800
same thing as a times
4a plus 1, right?

00:06:47.800 --> 00:06:50.490
When I factor an a out, I took
the a out of the parentheses,

00:06:50.490 --> 00:06:54.130
and then I divide each
of these terms by a.

00:06:54.130 --> 00:06:55.840
And that's going to
be equal to 0.

00:06:55.840 --> 00:07:01.190
So that means either a is equal
to 0, or 4a plus 1 is

00:07:01.190 --> 00:07:02.280
equal to 0.

00:07:02.280 --> 00:07:05.690
Well, that's one solution
right from the get go.

00:07:05.690 --> 00:07:06.910
And this one right
here, we could

00:07:06.910 --> 00:07:08.490
subtract 1 from both sides.

00:07:08.490 --> 00:07:11.090
We get 4a is equal
to negative 1.

00:07:11.090 --> 00:07:13.080
And then you could divide
both sides by 4.

00:07:13.080 --> 00:07:17.470
And we get a is equal
to negative 1/4.

00:07:17.470 --> 00:07:20.190
Now I said that was the last
one, let me do one more.

00:07:20.190 --> 00:07:22.870
Because I want to show you,
here, I'm factoring and then

00:07:22.870 --> 00:07:27.020
solving, but I want to show
you how useful a factored

00:07:27.020 --> 00:07:28.540
polynomial is.

00:07:28.540 --> 00:07:30.810
So I'm going to start with one
that's already factored.

00:07:30.810 --> 00:07:40.800
So say I had x minus 5 times 2x
plus 7 times 3x minus 4, is

00:07:40.800 --> 00:07:41.790
equal to 0.

00:07:41.790 --> 00:07:44.200
And if you're bored, you can
actually multiply these out

00:07:44.200 --> 00:07:48.830
and you'll find that you get
a third degree polynomial.

00:07:48.830 --> 00:07:51.040
But if you want to solve
that being equal to 0,

00:07:51.040 --> 00:07:52.190
it'd be very difficult.

00:07:52.190 --> 00:07:55.120
But this, we're going to
see, is not too bad.

00:07:55.120 --> 00:07:59.980
As we said, if we have three
numbers, if we have a times b

00:07:59.980 --> 00:08:03.970
times c equal to 0, that means
that at least one of these are

00:08:03.970 --> 00:08:05.170
0, maybe more than one.

00:08:05.170 --> 00:08:08.260
Either a or b or c, or some
combination of them,

00:08:08.260 --> 00:08:09.450
are equal to 0.

00:08:09.450 --> 00:08:11.170
So once again, I have
three numbers.

00:08:11.170 --> 00:08:13.480
This could be viewed as an a.

00:08:13.480 --> 00:08:15.310
This could be viewed as a b.

00:08:15.310 --> 00:08:17.070
This could be viewed as a c.

00:08:17.070 --> 00:08:19.730
I'm multiplying them and
they're equal to 0.

00:08:19.730 --> 00:08:23.500
So that means that either
x minus 5 is equal to 0.

00:08:23.500 --> 00:08:27.170
Or 2x plus 7 is equal to 0.

00:08:27.170 --> 00:08:31.010
Or 3x minus 4 is equal to 0.

00:08:31.010 --> 00:08:33.309
And we can solve each of
these independently.

00:08:33.309 --> 00:08:36.020
Add 5 to both sides of this
equation, and you get x is

00:08:36.020 --> 00:08:37.360
equal to 5.

00:08:37.360 --> 00:08:40.580
So that's one solution
to this equation.

00:08:40.580 --> 00:08:43.799
Over here, subtract
7 from both sides.

00:08:43.799 --> 00:08:46.560
You get 2x is equal
to negative 7.

00:08:46.560 --> 00:08:48.030
Divide both sides by 2.

00:08:48.030 --> 00:08:51.780
You get x is equal to
negative 7 over 2.

00:08:51.780 --> 00:08:53.570
That's another solution.

00:08:53.570 --> 00:08:56.870
And then finally, add 4 to both
sides of this equation.

00:08:56.870 --> 00:08:59.170
You get 3x is equal to 4.

00:08:59.170 --> 00:09:00.930
Divide both sides by 3.

00:09:00.930 --> 00:09:04.330
You get x is equal to 4/3.

00:09:04.330 --> 00:09:06.690
So this third degree polynomial,
if you were to

00:09:06.690 --> 00:09:09.760
multiply it out, actually had
three solutions to that

00:09:09.760 --> 00:09:12.320
polynomial equalling 0.

00:09:12.320 --> 00:09:14.530
And we're going to see, these
are actually called roots of

00:09:14.530 --> 00:09:18.260
the polynomial, the x values
that make that polynomial

00:09:18.260 --> 00:09:18.900
equal to 0.

00:09:18.900 --> 00:09:20.860
But hopefully, you've just
gotten an appreciation for

00:09:20.860 --> 00:09:24.040
one, how to factor a polynomial
in this video, and

00:09:24.040 --> 00:09:28.100
also, why it's useful for
solving polynomial equations.


Introduction to special products of binomials

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:02.770
And now I want to do a bunch
of examples dealing with

00:00:02.770 --> 00:00:07.160
probably the two most typical
types of polynomial

00:00:07.160 --> 00:00:10.550
multiplication that you'll see,
definitely, in algebra.

00:00:10.550 --> 00:00:13.640
And the first is just
squaring a binomial.

00:00:13.640 --> 00:00:20.600
So if I have x plus 9 squared, I
know that your temptation is

00:00:20.600 --> 00:00:24.630
going to say, oh, isn't that
x squared plus 9 squared?

00:00:24.630 --> 00:00:26.260
And I'll say, no, it isn't.

00:00:26.260 --> 00:00:28.260
You have to resist every
temptation on the

00:00:28.260 --> 00:00:29.660
planet to do this.

00:00:29.660 --> 00:00:32.280
It is not x squared
plus 9 squared.

00:00:32.280 --> 00:00:40.610
Remember, x plus 9 squared,
this is equal to x plus 9,

00:00:40.610 --> 00:00:43.630
times x plus 9.

00:00:43.630 --> 00:00:47.940
This is a multiplication of this
binomial times itself.

00:00:47.940 --> 00:00:49.350
You always need to
remember that.

00:00:49.350 --> 00:00:52.900
It's very tempting to think that
it's just x squared plus

00:00:52.900 --> 00:00:55.760
9 squared, but no, you have
to expand it out.

00:00:55.760 --> 00:00:59.160
And now that we've expanded it
out, we can use some of the

00:00:59.160 --> 00:01:02.250
skills we learned in the last
video to actually multiply it.

00:01:02.250 --> 00:01:04.370
And just to show you that we can
do it in the way that we

00:01:04.370 --> 00:01:09.910
multiplied the trinomial last
time, let's multiply x plus 9,

00:01:09.910 --> 00:01:15.760
times x plus a magenta 9.

00:01:15.760 --> 00:01:18.850
And I'm doing it this way just
to show you when I'm

00:01:18.850 --> 00:01:20.880
multiplying by this
9 versus this x.

00:01:20.880 --> 00:01:21.740
But let's just do it.

00:01:21.740 --> 00:01:23.970
So we go 9 times 9 is 81.

00:01:23.970 --> 00:01:25.800
Put it in the constants'
place.

00:01:25.800 --> 00:01:29.630
9 times x is 9x.

00:01:29.630 --> 00:01:32.830
Then we have-- go switch to this
x term-- we have a yellow

00:01:32.830 --> 00:01:36.240
x. x times 9x is 9x.

00:01:36.240 --> 00:01:38.022
Put it in the first
degree space.

00:01:38.022 --> 00:01:40.960
x times x is x squared.

00:01:43.670 --> 00:01:45.700
And then we add everything up.

00:01:45.700 --> 00:01:52.520
And we get x squared
plus 18x plus 81.

00:01:52.520 --> 00:01:57.650
So this is equal to x squared
plus 18x plus 81.

00:01:57.650 --> 00:02:00.560
Now you might see a little bit
of a pattern here, and I'll

00:02:00.560 --> 00:02:02.965
actually make the pattern
explicit in a second.

00:02:02.965 --> 00:02:06.010
But when you square a binomial,
what happened?

00:02:06.010 --> 00:02:07.750
You have x squared.

00:02:07.750 --> 00:02:11.190
You have this x times this
x, gives you x squared.

00:02:11.190 --> 00:02:13.920
You have the 9 times
the 9, which is 81.

00:02:13.920 --> 00:02:16.950
And then you have this term
here which is 18x.

00:02:16.950 --> 00:02:18.910
How did we get that 18x?

00:02:18.910 --> 00:02:23.210
Well, we multiplied this x times
9 to get 9x, and then we

00:02:23.210 --> 00:02:26.510
multiplied this 9 times
x to get another 9x.

00:02:26.510 --> 00:02:29.960
And then we added the two
right here to get 18x.

00:02:29.960 --> 00:02:33.250
So in general, whenever you have
a squared binomial-- let

00:02:33.250 --> 00:02:33.890
me do it this way.

00:02:33.890 --> 00:02:38.250
I'll do it in very general
terms. Let's say we have a

00:02:38.250 --> 00:02:39.630
plus b squared.

00:02:42.340 --> 00:02:44.740
Let me multiply it this way
again, just to give you the

00:02:44.740 --> 00:02:45.630
hang of it.

00:02:45.630 --> 00:02:53.250
This is equal to a plus b, times
a plus-- I'll do a green

00:02:53.250 --> 00:02:54.770
b right there.

00:02:54.770 --> 00:02:58.440
So we have to b times
b is b squared.

00:02:58.440 --> 00:03:00.410
Let's just assume that this
is a constant term.

00:03:00.410 --> 00:03:04.510
I'll put it in the b squared
right there.

00:03:04.510 --> 00:03:05.860
I'm assuming this is constant.

00:03:05.860 --> 00:03:07.060
So this would be a constant,
this would be

00:03:07.060 --> 00:03:08.660
analogous to our 81.

00:03:08.660 --> 00:03:11.640
a is a variable that we--
actually let me change that up

00:03:11.640 --> 00:03:13.040
even better.

00:03:13.040 --> 00:03:21.270
Let me make this into x plus b
squared, and we're assuming b

00:03:21.270 --> 00:03:22.750
is a constant.

00:03:22.750 --> 00:03:30.890
So it would be x plus b, times x
plus a green b, right there.

00:03:30.890 --> 00:03:34.700
So assuming b's a constant,
b times b is b squared.

00:03:34.700 --> 00:03:38.060
b times x is bx.

00:03:38.060 --> 00:03:40.090
And then we'll do
the magenta x.

00:03:40.090 --> 00:03:43.140
x times b is bx.

00:03:43.140 --> 00:03:46.900
And then x times
x is x squared.

00:03:46.900 --> 00:03:50.330
So when you add everything,
you're left with x squared

00:03:50.330 --> 00:03:56.840
plus 2bx, plus b squared.

00:03:56.840 --> 00:04:02.010
So what you see is, the end
product, what you have when

00:04:02.010 --> 00:04:06.340
you have x plus b squared, is
x squared, plus 2 times the

00:04:06.340 --> 00:04:10.490
product of x and b,
plus b squared.

00:04:10.490 --> 00:04:13.750
So given that pattern, let's
do a bunch more of these.

00:04:17.910 --> 00:04:19.709
And I'm going to do
it the fast way.

00:04:19.709 --> 00:04:23.850
So 3x minus 7 squared.

00:04:23.850 --> 00:04:25.980
Let's just remember
what I told you.

00:04:25.980 --> 00:04:29.340
Just don't remember it, in the
back of your mind, you should

00:04:29.340 --> 00:04:30.780
know why it makes sense.

00:04:30.780 --> 00:04:33.350
If I were to multiply this
out, do the distributive

00:04:33.350 --> 00:04:36.740
property twice, you know you'll
get the same answer.

00:04:36.740 --> 00:04:44.730
So this is going to be equal
to 3x squared, plus 2 times

00:04:44.730 --> 00:04:48.990
3x, times negative 7.

00:04:48.990 --> 00:04:49.280
Right?

00:04:49.280 --> 00:04:53.630
We know that it's 2 times each
the product of these terms,

00:04:53.630 --> 00:04:57.700
plus negative 7 squared.

00:04:57.700 --> 00:05:01.200
And if we use our product rules
here, 3x squared is the

00:05:01.200 --> 00:05:03.950
same thing as 9x squared.

00:05:03.950 --> 00:05:06.440
This right here, you're going
to have a 2 times a 3, which

00:05:06.440 --> 00:05:11.980
is 6, times a negative 7,
which is negative 42x.

00:05:11.980 --> 00:05:16.670
And then a negative 7
squared is plus 49.

00:05:16.670 --> 00:05:17.930
That was the fast way.

00:05:17.930 --> 00:05:20.600
And just to make sure that I'm
not doing something bizarre,

00:05:20.600 --> 00:05:22.950
let me do it the slow
way for you.

00:05:22.950 --> 00:05:32.320
3x minus 7, times 3x minus 7.

00:05:32.320 --> 00:05:36.940
Negative 7 times negative
7 is positive 49.

00:05:36.940 --> 00:05:42.310
Negative 7 times 3x
is negative 21x.

00:05:42.310 --> 00:05:47.220
3x times negative 7
is negative 21x.

00:05:47.220 --> 00:05:52.280
3x times 3x is 9 x squared.

00:05:52.280 --> 00:05:54.060
Scroll to the left
a little bit.

00:05:54.060 --> 00:05:55.050
Add everything.

00:05:55.050 --> 00:06:02.060
You're left with 9x squared,
minus 42x, plus 49.

00:06:02.060 --> 00:06:05.050
So we did indeed get
the same answer.

00:06:05.050 --> 00:06:07.580
Let's do one more, and we'll
do it the fast way.

00:06:07.580 --> 00:06:12.130
So if we have 8x minus 3--
actually, let me do one which

00:06:12.130 --> 00:06:13.920
has more variables in it.

00:06:13.920 --> 00:06:19.370
Let's say we had 4x squared plus
y squared, and we wanted

00:06:19.370 --> 00:06:21.360
to square that.

00:06:21.360 --> 00:06:22.720
Well, same idea.

00:06:22.720 --> 00:06:27.820
This is going to be equal to
this term squared, 4x squared,

00:06:27.820 --> 00:06:32.840
squared, plus 2 times the
product of both terms, 2 times

00:06:32.840 --> 00:06:39.210
4x squared times y
squared, plus y

00:06:39.210 --> 00:06:42.710
squared, this term, squared.

00:06:42.710 --> 00:06:44.800
And what's this going
to be equal to?

00:06:44.800 --> 00:06:48.430
This is going to be equal to
16-- right, 4 squared is 16--

00:06:48.430 --> 00:06:53.320
x squared, squared, that's 2
times 2, so it's x to the

00:06:53.320 --> 00:06:54.730
fourth power.

00:06:54.730 --> 00:06:58.330
And then plus, 2 times
4 times 1, that's

00:06:58.330 --> 00:07:01.820
8x squared y squared.

00:07:01.820 --> 00:07:07.300
And then y squared, squared,
is y to the fourth.

00:07:07.300 --> 00:07:10.390
Now, we've been dealing with
squaring a binomial.

00:07:10.390 --> 00:07:13.140
The next example I want to show
you is when I take the

00:07:13.140 --> 00:07:15.700
product of a sum and
a difference.

00:07:15.700 --> 00:07:18.350
And this one actually comes
out pretty neat.

00:07:18.350 --> 00:07:20.160
So I'm going to do a very
general one for you.

00:07:20.160 --> 00:07:28.160
Let's just do a plus
b, times a minus b.

00:07:28.160 --> 00:07:30.400
So what's this going
to be equal to?

00:07:30.400 --> 00:07:36.660
This is going to be equal to a
times a-- let me make these

00:07:36.660 --> 00:07:41.260
actually in different colors--
so a minus b, just like that.

00:07:41.260 --> 00:07:45.900
So it's going to be this green
a times this magenta a, a

00:07:45.900 --> 00:07:55.380
times a, plus, or maybe I should
say minus, the green a

00:07:55.380 --> 00:07:56.670
times this b.

00:07:56.670 --> 00:07:59.560
I got the minus from
right there.

00:07:59.560 --> 00:08:04.370
And then we're going to have the
green b, so plus the green

00:08:04.370 --> 00:08:07.230
b times the magenta a.

00:08:07.230 --> 00:08:10.920
I'm just multiplying every
term by every term.

00:08:10.920 --> 00:08:17.250
And then finally minus the green
b-- that's where the

00:08:17.250 --> 00:08:19.050
minus is coming from--
minus the green b

00:08:19.050 --> 00:08:22.370
times the magenta b.

00:08:22.370 --> 00:08:23.590
And what is this going
to be equal to?

00:08:23.590 --> 00:08:27.220
This is going to be equal
to a squared, and then

00:08:27.220 --> 00:08:29.020
this is minus ab.

00:08:29.020 --> 00:08:32.840
This could be rewritten as
plus ab, and then we have

00:08:32.840 --> 00:08:35.039
minus b squared.

00:08:35.039 --> 00:08:39.130
These right here cancel out,
minus ab plus ab, so you're

00:08:39.130 --> 00:08:42.600
just left with a squared
minus b squared.

00:08:42.600 --> 00:08:44.430
Which is a really neat
result because it

00:08:44.430 --> 00:08:46.120
really simplifies things.

00:08:46.120 --> 00:08:49.190
So let's use that notion to
do some multiplication.

00:08:49.190 --> 00:08:55.470
So if we say 2x minus
1, times 2x plus 1.

00:08:55.470 --> 00:08:56.710
Well, these are the
same thing.

00:08:56.710 --> 00:09:01.810
The 2x plus 1, you could view
this as, if you like, a plus

00:09:01.810 --> 00:09:06.750
b, and the 2x minus 1, you can
view it as a minus b, where

00:09:06.750 --> 00:09:10.830
this is a, and that b is 1.

00:09:10.830 --> 00:09:13.440
This is b.

00:09:13.440 --> 00:09:15.010
That is a.

00:09:15.010 --> 00:09:18.660
Just using this pattern that
we figured out just now.

00:09:18.660 --> 00:09:19.680
So what is this going
to be equal to?

00:09:19.680 --> 00:09:24.820
It's going to be a squared, it's
going to be 2x squared,

00:09:24.820 --> 00:09:29.490
minus b squared, minus
1 squared.

00:09:29.490 --> 00:09:32.790
2x squared is 4x squared.

00:09:32.790 --> 00:09:36.180
1 squared is just
1, so minus 1.

00:09:36.180 --> 00:09:39.210
So it's going to be 4x
squared minus 1.

00:09:39.210 --> 00:09:41.560
Let's do one more of these,
just to really

00:09:41.560 --> 00:09:42.890
hit the point home.

00:09:45.590 --> 00:09:47.290
I'll just focus on
multiplication right now.

00:09:47.290 --> 00:09:52.040
If I have 5a minus 2b, and
I'm multiplying that

00:09:52.040 --> 00:09:54.910
times 5a plus 2b.

00:09:54.910 --> 00:09:58.640
And remember, this only applies
when I have at a

00:09:58.640 --> 00:10:00.750
product of a sum and
a difference.

00:10:00.750 --> 00:10:02.410
That's the only time that
I can use this.

00:10:02.410 --> 00:10:03.850
And I've shown you why.

00:10:03.850 --> 00:10:06.320
And if you're ever in doubt,
just multiply it out.

00:10:06.320 --> 00:10:07.880
It'll take you a little
bit longer.

00:10:07.880 --> 00:10:09.630
And you'll see the terms
canceling out.

00:10:09.630 --> 00:10:13.070
You can't do this for just any
binomial multiplication.

00:10:13.070 --> 00:10:14.590
You saw that earlier in the
video, when we were

00:10:14.590 --> 00:10:17.080
multiplying, when we were
taking squares.

00:10:17.080 --> 00:10:19.560
So this is going to be, using
the pattern, it's going to be

00:10:19.560 --> 00:10:28.310
5a squared minus 2b squared,
which is equal to 25 a squared

00:10:28.310 --> 00:10:31.400
minus 4b squared.

00:10:31.400 --> 00:10:34.330
And, well, I'll leave it there,
and I'll see you in the

00:10:34.330 --> 00:10:35.900
next video.