Example 1: Multiplying a binomial by a binomial

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Language: en

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Multiply (3x+2) by (5x-7). So we are multiplying two binomials. I am actually going to show you

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two really equivalent ways of doing this. One that you might hear in a classroom and it is kind of a more mechanical

00:00:13.600 --> 00:00:18.533
memorizing way of doing it which might be faster but you really don't know what you are doing

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and then there is the one where you are essentially just applying something what you already know

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and kind of a logical way. So I will first do the memorizing way that you might be exposed to and they'll use something called FOIL.

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So let me write this down here. So you can immediately see that whenever someone gives you

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a new mnemonic to memorize, that you are doing something pretty mechanical. So FOIL literally stands for

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First Outside, let me write it this way.....F O I L where the F in FOIL stands for First, the O in FOIL stand for Outside,

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the I stands for Inside and then the L stands for Last. The reason why I don't like these things

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is that when you are 35 years old, you are not going to remember what FOIL stood for and

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then you are not going to remember how to multiply this binomial. But lets just apply FOIL.

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So First says just multiply the first terms in each of these binomials. So just multiply the 3x times the 5x.

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So (3x. 5x). The Outside part tells us to multiply the outside terms. So in this case, you have

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3x on the outside and you have -7 on the outside. So that is +3x(-7). The inside,

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well the inside terms here are 2 and 5x. So, (+2.5x) and then finally you have the last terms. You have the 2 and the -7.

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So the last terms are 2 times -7. 2(-7). So what you are essentially doing is just making sure

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that you are multipying each term by every other term here. What we are essentially doing is multiplying,

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doing the distributive property twice. We are multiplying the 3x times (5x-7). So 3x times (5x-7) is (3x . 5x) plus (3x - 7).

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And we are multiplying the 2 times (5x-7) to give us these terms.

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But anyway, lets just multiply these out just to get our answer. 3x times 5x is same thing as (3 times 5) ( x times x)

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which is the same thing as 15x square. You can just do this x to the first time to x to the first.

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You multiply the x to get x squared. 3 times 5 is 15. This term right here 3 times -7 is -21

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and then you have your x right over here. And then you have this term which is 2 times 5 which is 10 times x. So +10x.

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And then finally you have this term here in blue. 2 times -7 is -14. And we aren't done yet,

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we can simplify this a little bit. We have two like terms here. We have this...let me find a new color.

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We have 2 terms with a x to the first power or just an x term right over here. So we have -21 of something and you add 10

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or in another way, you have 10 of something and you subtract 21 of them, you are going to have -11 of that something.

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We put the other terms here, you have 15... 15x squared and then you have your -14 and we are done.

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Now I said I would show you another way to do it. I want to show you why the

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distributive property can get us here without having to memorize FOIL.

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So the distributive property tells us that if we 're... look if we are multipying something times an expression, you just have to multiply

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times every term in the expression. So we can distribute, we can distribute the 5x onto the 3...,

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or actually we could...well, let me view it this way... we could distribute the 5x-7, this whole thing onto the 3x+2.

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Let me just change the order since we are used to distributing something from the left.

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So this is the same thing as (5x-7)(3x+2). I just swapped the two expressions. And we can distribute this whole thing

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times each of these terms. Now what happens if I take (5x-7) times 3x? Well, thats just going

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to be 3x times (5x-7). So I have just distributed the 5x-7 times 3x and to that I am going to add 2 times 5x-7.

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I have just distributed the 5x-7 onto the 2.

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Now, you can do the distributive property again. We can distribute the

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3x onto the 5x. We can distribute the 3x onto the 5x.

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And we can distribute the 3x onto the -7.

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We can distribute the 2 onto the 5x, over here

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and we can distribute the 2 on that -7.

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Now if we do it like this what do we get ?

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3x times 5x, that's this right over here. If we do 3x times -7, that's this term right over here.

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If you do 2 times 5x, that's this term right over here. If you do 2 times -7, that is this term right

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over here. So we got the exact same result that we got with FOIL.

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Now, FOIL can be faster if you just wanted to do it and kind of skip to this step.

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I think its important that you know that this is how it actually works.

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Just in case you do forget this when you are 35 or 45 years old and you are faced with multiplying binomial,

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you just have to remember the distributive property.


Example 1: Multiplying two monomials with a single variable

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Language: en

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Express the area of a rectangle
with the length 4y and width

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2y as a monomial.

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So the area of a
rectangle is just

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the width times the height,
or the base times the height.

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Now here, the base is 4y.

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So let me just
write it down-- area

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is going to be
equal to the width

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of the base times the height.

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The width is 4y and
the height here is 2y.

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So the area is going
to be 4y times 2y.

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Now here we can just
use the associative

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and the commutative
properties of multiplication.

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So just swap the order that
we do this multiplying in.

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Instead of multiplying 4
times y times 2 times y,

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we could say this is the same
thing as 4 times 2 times y

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times y.

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And of course, 4 times 2,
that part right over there

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is just equal to 8.

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And then y times y, well
that's just y squared.

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Or it's y to the first
times y to the first,

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which is also y squared.

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So it becomes 8y squared.

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So that's the area
and it's a monomial.

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This is a polynomial
with only one term.


Multiplying polynomials example

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Language: en

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Multiply (5a - 2)(4a^2 + 3a - 1)

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So here we are multiplying a binomial by a trinomial

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So the FOIL tool will not work here

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It would only work if you were multiplying a binomial by another binomial

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So here we really have to rely on the distributive property. What we can do is

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we can distribute this entire trinomial onto the binomial. We we can multiply 4a^2

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plus 3a -1 times 5a and then multiply (4a^2 + 3a -1) times -2.

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So that would give us, let me do it this way. That would give us 5a times 4a^2 +3a -1

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and then we would have a plus -2 times 4a^2 plus 3a -1. I've just distributed this

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onto each term over here you see the 5a -2 each term is now being multiplied by this entire thing.

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And now we can distribute the 5a onto this term. So we have 5a times 4a^2.

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So we can multiply the 5 times the 4 we get 20. a times the a^2 is a^3. Then we do

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5a times 3a. 5 times 3 is fifteen. a times a is a^2.

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Then we have 5a times -1, well thats just going to be -5a. Now we do this part,

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Plus -2 times 4a^2. -2 times four is -8a^2. So we just did that...

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-2 times 3a is -6a. And then finally you have -2 times -1. Well a negative times a negative is a positive.

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So that is postive 2 and now we can combine like terms.

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We only have one ^3 term over here. We only have the 20a^3. So lets just write that down.

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And then lets look at our a^2 terms we have a 15a^2 right over here and we have another

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-8a^2. So if I have 15 of something and I subtract 8 from that then I have 7 of that something left.

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So 15 minus 8 is 7. And then we have a -5a. And to that we're going to add -6a.

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So we are already -5 and to that we're going to be another -6.

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We going to be 6 more negative. So that is -11a. And then finally we have this constant out here.

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We have this plus 2. No I'll show you another way to get this exact same answer.

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We are done. We have simplified, and we have combined all our like terms.

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Another way to do this is just like doing the distributive property twice. But it's essentially

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more analogous to how we multiply numbers. So if we do it like this we could take this

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expression up here. We could take 4a^2 plus 3a -1. And then multiply it by this second expression.

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So times.... and I'll write it in the same places.. and when I say places. I'm not saying

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powers of ten, I'm saying powers of a. So this is in the first a power of a space.

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So 5a, see we have first powers of a in this space. Minus 2 this is the zeroth power of a.

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This a to zeroth place instead of the ones place. This is the a to the first power

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instead of the tens place. And this is a to the square power instead of the hundreds place.

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Completely analogous to what you do when you learn to multiply numbers in the first

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or second grade. So let's multiply these. First we'll multiply -2 by this entire thing.

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Remember we're just essentially doing the distributive property. When I multiply -2 by this entire thing

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I'm doing this step right over here. So -2 times -1 is positive 2. And once again theres no a there.

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So I'm going to write this in the a^0 place or the ones place.

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So it's going to be positive 2. -2 times 3a. Well that is -6a. And it's going to be in the a place.

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-6a, a to the first power place. And then -2 times 4a^2, well that's just going to be -8a^2.

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So this is-2 being multiplied by all of that gives me this. And then when I take my 5a and I multiply

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it by -1 that going to be -5a. That's going to the a place, that goes under a like term.

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then 5a times 3a, 5 times 3 is 15 and a times a is a^2. So it goes under the other a^2.

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So it's 15a^2 and then finally 5a times 4a^2 . 5 times 4 is 20. a times a^2 is a^3. So you have 20a^3.

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And now you can just add it all up. And we get 2 plus nothing is 2. -6a -5a well that's just -11a

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We still have this plus 2 here. -8a^2 plus 15a^2 well thats going to be 7a^2.

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So plus 7a^2. And then you're adding this to nothing so you have 20a^3. And we got the exact same answer.

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Cause we really did the same exact thing we just wrote it in a different way.


Polynomial divided by monomial

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Language: en

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Simplify the expression
18x to the fourth minus 3x

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squared plus 6x minus
4, all of that over 6x.

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So there's a couple ways
to think about them.

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They're all really equivalent.

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You can really just
view this up here

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as being the exact same thing
as 18x to the fourth over 6x

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plus negative 3x
squared over 6x,

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or you could say minus
3x squared over 6x,

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plus 6x over 6x,
minus 4 over 6x.

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Now, there's a couple of
ways to think about it.

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One is I just kind of decomposed
this numerator up here.

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If I just had a bunch
of stuff, a plus b

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plus c over d, that's clearly
equal to a/d plus b/d plus c/d.

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Or maybe not so clearly,
but hopefully that helps

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clarify it.

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Another way to think
about it is kind

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of like you're
distributing the division.

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If I divide a whole
expression by something,

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that's equivalent to
dividing each of the terms

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by that something.

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The other way to
think about it is

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that we're multiplying
this entire expression.

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So this is the same
thing as 1 over 6x times

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this entire thing, times
18x to the fourth minus

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3x squared plus 6x minus 4.

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And so here, this would just
be the straight distributive

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property to get to this.

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Whatever seems to
make sense for you--

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they're are all equivalent.

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They're all logical, good things
to do to simplify this thing.

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Now, once you have
it here, now we just

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have a bunch of monomials that
we're just dividing by 6x.

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And here, we could just
use exponent properties.

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This first one over here,
we can take the coefficients

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and divide them.

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18 divided by 6 is 3.

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And then you have x
to the fourth divided

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by x to the-- well,
they don't tell us.

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But if it's just an x,
that's the same thing

00:01:53.840 --> 00:01:55.310
as x to the first power.

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So it's x to the fourth
divided by x to the first.

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That's going to be x to
the 4 minus 1 power, or x

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to the third power.

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Then we have this
coefficient over here,

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or these coefficients.

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We have negative 3 divided by 6.

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So I'm going to
do this part next.

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Negative 3 divided
by 6 is negative 1/2.

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And then you have x
squared divided by x.

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We already know that x is the
same thing as x to the first.

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So that's going to be x to the 2
minus 1 power, which is just 1.

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Or I could just leave
it as an x right there.

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Then we have these
coefficients, 6 divided by 6.

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Well, that's just 1.

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So I could just--
well, I'll write it.

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I could write a 1 here.

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And let me just write the 1
here, because we said 2 minus 1

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is 1.

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And then x divided by x is x to
the first over x to the first.

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You could view it two ways--
anything divided by anything

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is just 1.

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Or you could view it as x to
the 1 divided by x to the 1

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is going to be x
to the 1 minus 1,

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which is x to the 0,
which is also equal to 1.

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Either way, you knew how
to do this before you even

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learned that exponent property.

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Because x divided by x
is 1, and then assuming

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x does not equal to 0.

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And we kind of have to
assume x doesn't equal 0

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in this whole thing.

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Otherwise, we would
be dividing by 0.

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And then finally,
we have 4 over 6x.

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And there's a couple of
ways to think about it.

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So the simplest way
is negative 4 over 6

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is the same thing
as negative 2/3.

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Just simplified that fraction.

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And we're multiplying
that times 1/x.

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So we can view this 4 times 1/x.

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Another way to think
about it is you

00:03:45.970 --> 00:03:49.230
could have viewed this 4 as
being multiplied by x to the 0

00:03:49.230 --> 00:03:52.610
power, and this being
x to the first power.

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And then when you tried to
simplify it using your exponent

00:03:55.710 --> 00:03:59.240
properties, you would have--
well, that would be x to the 0

00:03:59.240 --> 00:04:02.772
minus 1 power, which is x
to the negative 1 power.

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So we could have written
an x to the negative 1

00:04:04.730 --> 00:04:08.370
here, but x to the negative 1
is the exact same thing as 1

00:04:08.370 --> 00:04:09.490
over x.

00:04:09.490 --> 00:04:11.950
And so let's just write our
answer completely simplified.

00:04:11.950 --> 00:04:18.220
So it's going to be 3x to the
third minus 1/2 x plus 1--

00:04:18.220 --> 00:04:22.960
because this thing
right here is just 1--

00:04:22.960 --> 00:04:26.540
and then minus 2 times
1 in the numerator

00:04:26.540 --> 00:04:30.220
over 3 times x in
the denominator.

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And we are done.

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Or we could write this.

00:04:32.150 --> 00:04:34.260
Depending on what you
consider more simplified,

00:04:34.260 --> 00:04:35.760
this last term right
here could also

00:04:35.760 --> 00:04:39.360
be written in minus 2/3
x to the negative 1.

00:04:39.360 --> 00:04:41.160
But if you don't want
a negative exponent,

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you could write it like that.


Simplifying rational expressions (old video)

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Kind: captions
Language: en

00:00:00.640 --> 00:00:07.150
Simplify the expression 28b
to the sixth divided by 7b.

00:00:07.150 --> 00:00:13.540
Well this is the same
thing as 28/7 times

00:00:13.540 --> 00:00:15.480
b to the sixth over b.

00:00:15.480 --> 00:00:17.440
And I did it this
way so that we have

00:00:17.440 --> 00:00:21.870
b to the sixth over b
to some other power.

00:00:21.870 --> 00:00:25.170
And this is over b, which is the
same thing as b to the first.

00:00:25.170 --> 00:00:28.200
So we could simplify this
using exponent properties.

00:00:28.200 --> 00:00:31.430
And 28/7 we can just
clearly simplify because 28

00:00:31.430 --> 00:00:33.122
is divisible by 7.

00:00:33.122 --> 00:00:36.920
28 divided by 7 is just 4.

00:00:36.920 --> 00:00:39.120
So that is just 4.

00:00:39.120 --> 00:00:41.260
And then b to the
sixth divided by b

00:00:41.260 --> 00:00:45.310
to the one, that's just going
to be b to the sixth minus 1

00:00:45.310 --> 00:00:45.860
power.

00:00:45.860 --> 00:00:50.830
So times b to the sixth
minus 1 or b to the fifth.

00:00:50.830 --> 00:00:55.810
So our answer is
4b to the fifth.


Example 1: Multiplying binomials to get a difference of squares

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:04.740
Find the product 2x
plus 8 times 2x minus 8.

00:00:04.740 --> 00:00:06.650
So we're multiplying
two binomials.

00:00:06.650 --> 00:00:09.080
So you could use FOIL,
you could just straight up

00:00:09.080 --> 00:00:10.675
use the distributive
property here.

00:00:10.675 --> 00:00:13.020
But the whole point of
this problem, I'm guessing,

00:00:13.020 --> 00:00:15.140
is to see whether you
recognize a pattern here.

00:00:15.140 --> 00:00:21.580
This is of the form a
plus b times a minus

00:00:21.580 --> 00:00:25.060
b, where here a
is 2x and b is 8.

00:00:25.060 --> 00:00:31.592
We have 2x plus 8 and then 2x
minus 8. a plus b, a minus b.

00:00:31.592 --> 00:00:33.050
What I want to do
is I'm just going

00:00:33.050 --> 00:00:34.929
to multiply this out for us.

00:00:34.929 --> 00:00:36.220
And then just see what happens.

00:00:36.220 --> 00:00:39.040
Whenever you have this pattern,
what the product actually looks

00:00:39.040 --> 00:00:39.790
like.

00:00:39.790 --> 00:00:41.450
So if you were to
multiply this out,

00:00:41.450 --> 00:00:43.990
we can distribute the a plus b.

00:00:43.990 --> 00:00:46.340
We could distribute
this whole thing.

00:00:46.340 --> 00:00:48.645
Distribute the whole
a plus b on the a

00:00:48.645 --> 00:00:50.490
and then distribute it on the b.

00:00:50.490 --> 00:00:52.170
And I could have done this
with this problem right here,

00:00:52.170 --> 00:00:54.420
and it would have taken us
less time to just solve it.

00:00:54.420 --> 00:00:57.150
But I want to find out
the general pattern here.

00:00:57.150 --> 00:00:59.240
So a plus b times a.

00:00:59.240 --> 00:01:04.030
So we have a times a plus
b, that's this times this.

00:01:04.030 --> 00:01:06.640
And then a plus b
times negative b,

00:01:06.640 --> 00:01:10.440
that's negative
b times a plus b.

00:01:10.440 --> 00:01:12.470
So I've done distributive
property once,

00:01:12.470 --> 00:01:13.890
now I could do it again.

00:01:13.890 --> 00:01:17.550
I can distribute the a
onto the a and this b

00:01:17.550 --> 00:01:19.170
and it gives me a squared.

00:01:19.170 --> 00:01:24.600
a times a is a squared,
plus a times b, which is ab.

00:01:24.600 --> 00:01:27.400
And now I can do it
with the negative b.

00:01:27.400 --> 00:01:34.160
Negative b times a is negative
ab or negative ba, same thing.

00:01:34.160 --> 00:01:38.980
And negative b times b
is negative b squared.

00:01:38.980 --> 00:01:40.490
Now, what does this simplify to?

00:01:40.490 --> 00:01:42.970
Well, I have an ab, and
then I'm subtracting an ab.

00:01:42.970 --> 00:01:46.190
So these two guys
cancel out and I am just

00:01:46.190 --> 00:01:49.530
left with a squared
minus b squared.

00:01:49.530 --> 00:01:51.600
So the general pattern,
and this is a good one

00:01:51.600 --> 00:01:54.620
to just kind of
know super fast, is

00:01:54.620 --> 00:01:57.040
that a plus b times
a minus b is always

00:01:57.040 --> 00:01:59.150
going to be a squared
minus b squared.

00:01:59.150 --> 00:02:03.490
So we have an a plus
b times an a minus b.

00:02:03.490 --> 00:02:06.920
So this product is
going to be a squared.

00:02:06.920 --> 00:02:11.360
So it's going to be
2x squared minus b

00:02:11.360 --> 00:02:15.250
squared minus 8 squared.

00:02:15.250 --> 00:02:16.760
2x squared, that's
the same thing

00:02:16.760 --> 00:02:21.850
as 2 squared times x
squared, or 4x squared.

00:02:21.850 --> 00:02:23.910
And from that, we're
subtracting 8 squared.

00:02:23.910 --> 00:02:28.250
So it's going to be
4x squared minus 64.


Example 2: Finding the square of a binomial with one variable

WEBVTT
Kind: captions
Language: en

00:00:00.102 --> 00:00:02.438
We're asked to simply, or expand

00:00:02.438 --> 00:00:05.875
(7x + 10) ^ 2

00:00:05.875 --> 00:00:07.872
Now the first thing I will show you

00:00:07.872 --> 00:00:10.008
is exactly what you should NOT do,

00:00:10.008 --> 00:00:11.819
well there's this huge temptation.

00:00:11.819 --> 00:00:13.444
A lot of people will look at this and say

00:00:13.444 --> 00:00:17.438
oh, that's (7x)^2 + 10^2.

00:00:17.438 --> 00:00:19.900
This is WRONG.

00:00:19.900 --> 00:00:22.082
And I'll write it in caps.

00:00:22.082 --> 00:00:24.311
This is WRONG!

00:00:24.311 --> 00:00:25.844
What your brain is doing is thinking

00:00:25.844 --> 00:00:29.838
if I had 7x times 10 and I squared that,

00:00:29.838 --> 00:00:35.689
this would be (7x)^2 times 10^2.

00:00:35.689 --> 00:00:38.893
We aren't multiplying here, we're adding 7x to 10.

00:00:38.893 --> 00:00:41.726
So you can't just square each of these terms.

00:00:41.726 --> 00:00:43.398
I just wanted to highlight,

00:00:43.398 --> 00:00:46.788
this is completely wrong,

00:00:46.788 --> 00:00:49.250
and to see why it's wrong,

00:00:49.250 --> 00:00:50.689
you have to remind yourself

00:00:50.689 --> 00:00:54.637
that (7x + 10)^2 is the exact same thing

00:00:54.637 --> 00:01:00.767
as (7x + 10)(7x + 10).

00:01:00.767 --> 00:01:02.624
That's what it means to square something.

00:01:02.624 --> 00:01:08.058
You're multiplying it by itself twice here.

00:01:08.058 --> 00:01:09.451
So this is what it is,

00:01:09.451 --> 00:01:14.095
so we're really just multiplying

00:01:14.095 --> 00:01:16.278
a binomial, or two binomials,

00:01:16.278 --> 00:01:17.857
they just happen to be the same one,

00:01:17.857 --> 00:01:19.203
and you could use F.O.I.L.,

00:01:19.203 --> 00:01:20.782
you could use the distributive method,

00:01:20.782 --> 00:01:22.268
but this is actually a special case:

00:01:22.268 --> 00:01:23.754
when you're squaring a binomial,

00:01:23.754 --> 00:01:25.055
so let's just think about it

00:01:25.055 --> 00:01:26.541
as a special case first then we can

00:01:26.541 --> 00:01:28.677
apply whatever we learn to this.

00:01:28.677 --> 00:01:30.256
So we could've just done it straight here,

00:01:30.256 --> 00:01:31.881
but I want to learn the general case

00:01:31.881 --> 00:01:33.367
so you can apply it to any problem

00:01:33.367 --> 00:01:34.854
that you might see.

00:01:34.854 --> 00:01:38.104
If I have (a+b) squared

00:01:38.104 --> 00:01:40.751
We already realised that it's not a squared plus b squared

00:01:40.751 --> 00:01:45.535
That is a plus b times a plus b.

00:01:45.535 --> 00:01:48.785
and now we can use the distributive property

00:01:48.785 --> 00:01:51.618
We can distribute this a + b times this a

00:01:51.618 --> 00:01:56.495
So we get, we get a times a plus b

00:01:56.495 --> 00:01:59.188
and we can distribute the a plus b times this b

00:01:59.188 --> 00:02:04.296
plus b times a plus b, and we distribute this a

00:02:04.296 --> 00:02:09.683
we get a squared plus ab plus b times a is another ab

00:02:09.683 --> 00:02:12.516
And I'm just swapping the order so it's the same as this.

00:02:12.516 --> 00:02:15.488
plus b times b which is b squared.

00:02:15.488 --> 00:02:18.553
These are the same or these are like terms.

00:02:18.553 --> 00:02:21.015
So we can add them. One of something plus another of

00:02:21.015 --> 00:02:24.033
that something will give you two of that something. 2 ab.

00:02:24.033 --> 00:02:27.609
We have a squared plus 2 ab plus b squared.

00:02:27.609 --> 00:02:32.346
So the pattern here, the pattern here, if I have a plus b squared

00:02:32.346 --> 00:02:37.176
it's equal to a squared plus 2 times the product of these numbers

00:02:37.176 --> 00:02:42.238
plus b squared. So over here I have seven x plus ten squared

00:02:42.238 --> 00:02:45.674
So this is going to be equal to seven x squared

00:02:45.674 --> 00:02:50.690
seven x squared plus 2 times the product of seven x and 10.

00:02:50.690 --> 00:02:55.752
2 times seven x times 10 plus 10 squared.

00:02:55.752 --> 00:02:59.374
So, the difference between the right answer and the wrong answer

00:02:59.374 --> 00:03:02.718
is that you have this middle term here that you might have

00:03:02.718 --> 00:03:04.854
forgotten about if you did it this way.

00:03:04.854 --> 00:03:07.780
And this comes out when you are multiplying all the different

00:03:07.780 --> 00:03:09.219
combinations of the terms here.

00:03:09.219 --> 00:03:12.563
if we simplify this, if we simplify seven x squared

00:03:12.563 --> 00:03:15.024
That's seven squared times x squared.

00:03:15.024 --> 00:03:18.136
So seven squared is 49 times x squared

00:03:18.136 --> 00:03:22.733
When you multiply this part out 2 times 7 times 10

00:03:22.733 --> 00:03:28.353
which is 140 and then we have our x.

00:03:28.353 --> 00:03:32.346
No other x there. And then plus 10 squared.

00:03:32.346 --> 99:59:59.999
So plus 100. And we are done.


Dividing polynomials 1

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:05.620
Divide x squared minus 3x
plus 2 divided by x minus 2.

00:00:05.620 --> 00:00:07.580
So we're going to
divide this into that.

00:00:07.580 --> 00:00:09.880
And we can do this really
the same way that you first

00:00:09.880 --> 00:00:10.796
learned long division.

00:00:10.796 --> 00:00:13.930
So we have x minus
2 being divided

00:00:13.930 --> 00:00:17.625
into x squared minus 3x plus 2.

00:00:17.625 --> 00:00:19.000
Another way we
could have written

00:00:19.000 --> 00:00:24.250
the same exact expression is
x squared minus 3x plus 2,

00:00:24.250 --> 00:00:26.520
all of that over x minus 2.

00:00:26.520 --> 00:00:29.710
That, that, and that are
all equivalent expressions.

00:00:29.710 --> 00:00:32.549
Now, to do this type
of long division--

00:00:32.549 --> 00:00:34.300
we can call it algebraic
long division--

00:00:34.300 --> 00:00:36.800
you want to look at the highest
degree term on the x minus 2

00:00:36.800 --> 00:00:39.890
and the highest degree term on
the x squared minus 3x plus 2.

00:00:39.890 --> 00:00:42.250
And here's the x, and
here's the x squared.

00:00:42.250 --> 00:00:44.590
x goes into x squared
how many times?

00:00:44.590 --> 00:00:47.160
Or x squared divided
by x is what?

00:00:47.160 --> 00:00:49.270
Well, that's just equal to x.

00:00:49.270 --> 00:00:51.254
So x goes into x
squared x times.

00:00:51.254 --> 00:00:53.170
And I'm going to write
it in this column right

00:00:53.170 --> 00:00:55.870
here above all of the x terms.

00:00:55.870 --> 00:00:58.870
And then we want to
multiply x times x minus 2.

00:00:58.870 --> 00:01:02.770
That gives us-- x
times x is x squared.

00:01:02.770 --> 00:01:05.770
x times negative
2 is negative 2x.

00:01:05.770 --> 00:01:08.470
And just like you first
learned in long division,

00:01:08.470 --> 00:01:09.970
you want to subtract
this from that.

00:01:13.270 --> 00:01:15.970
But that's completely the
same as adding the opposite,

00:01:15.970 --> 00:01:18.670
or multiplying each of
these terms by negative 1

00:01:18.670 --> 00:01:19.570
and then adding.

00:01:19.570 --> 00:01:21.370
So let's multiply
that times negative 1.

00:01:21.370 --> 00:01:24.670
And negative 2x times
negative 1 is positive 2x.

00:01:24.670 --> 00:01:25.870
And now let's add.

00:01:25.870 --> 00:01:28.270
x squared minus x
squared-- those cancel out.

00:01:28.270 --> 00:01:31.870
Negative 3x plus 2x--
that is negative x.

00:01:31.870 --> 00:01:33.970
And then we can bring
down this 2 over here.

00:01:33.970 --> 00:01:38.770
So it's negative x plus 2 left
over, when we only go x times.

00:01:38.770 --> 00:01:41.770
So then we say, can x minus
2 go into negative x plus 2?

00:01:41.770 --> 00:01:46.520
Well, x goes into negative
x negative one times.

00:01:46.520 --> 00:01:47.770
You can look at it right here.

00:01:47.770 --> 00:01:50.470
Negative x divided
by x is negative 1.

00:01:50.470 --> 00:01:52.270
These guys cancel out.

00:01:52.270 --> 00:01:54.370
Those guys cancel out.

00:01:54.370 --> 00:01:57.370
So negative 1 times x minus
2-- you have negative 1

00:01:57.370 --> 00:01:59.470
times x, which is negative x.

00:01:59.470 --> 00:02:03.970
Negative 1 times
negative 2 is positive 2.

00:02:03.970 --> 00:02:06.970
And we want to subtract
this from that,

00:02:06.970 --> 00:02:08.770
just like you do
in long division.

00:02:08.770 --> 00:02:11.770
But that's the same thing
as adding the opposite,

00:02:11.770 --> 00:02:14.104
or multiplying each of
these terms by negative 1

00:02:14.104 --> 00:02:14.770
and then adding.

00:02:14.770 --> 00:02:18.970
So negative x times
negative 1 is positive x.

00:02:18.970 --> 00:02:23.770
Positive 2 times
negative 1 is negative 2.

00:02:23.770 --> 00:02:26.170
These guys cancel
out, add up to 0.

00:02:26.170 --> 00:02:27.370
These guys add up to 0.

00:02:27.370 --> 00:02:29.770
We have no remainder.

00:02:29.770 --> 00:02:34.270
So we got this as being
equal to x minus 1.

00:02:34.270 --> 00:02:35.470
And we can verify it.

00:02:35.470 --> 00:02:41.170
If we multiply x minus 1 times
x minus 2, we should get this.

00:02:41.170 --> 00:02:42.670
So let's actually do that.

00:02:42.670 --> 00:02:48.670
So let's multiply x
minus 1 times x minus 2.

00:02:48.670 --> 00:02:51.670
So let's multiply negative
2 times negative 1.

00:02:51.670 --> 00:02:53.770
That gives us positive 2.

00:02:53.770 --> 00:02:57.370
Negative 2 times x--
that's negative 2x.

00:02:57.370 --> 00:03:00.070
Let's multiply x
times negative 1.

00:03:00.070 --> 00:03:02.170
That is negative x.

00:03:02.170 --> 00:03:06.070
And then x times x is x squared.

00:03:06.070 --> 00:03:07.870
And then add all the like terms.

00:03:07.870 --> 00:03:12.370
x squared, negative 2x minus
x-- that's negative 3x.

00:03:12.370 --> 00:03:15.670
And then 2 plus
nothing is just 2.

00:03:15.670 --> 00:03:19.400
And so we got that
polynomial again.


Dividing polynomials with remainders

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:03.970
We divide x to the
third plus 5x minus 4

00:00:03.970 --> 00:00:07.600
by x minus x--
actually, I think this

00:00:07.600 --> 00:00:09.540
is supposed to be an x squared.

00:00:09.540 --> 00:00:10.290
Let me correct it.

00:00:10.290 --> 00:00:13.690
So we're going to divide x
to the third plus 5x minus 4

00:00:13.690 --> 00:00:18.390
by x squared minus x plus 1.

00:00:18.390 --> 00:00:20.810
And just to see the different
ways we can rewrite this.

00:00:20.810 --> 00:00:23.700
We could rewrite this
as x to the third plus

00:00:23.700 --> 00:00:31.090
5x minus 4 divided by x
squared minus x plus 1.

00:00:31.090 --> 00:00:33.130
Or maybe the best
way to write it

00:00:33.130 --> 00:00:34.650
in this circumstance,
since we're

00:00:34.650 --> 00:00:36.590
going to do algebraic
long division,

00:00:36.590 --> 00:00:42.140
is to write it as x
squared minus x plus 1,

00:00:42.140 --> 00:00:47.099
divided into x to
the third plus--

00:00:47.099 --> 00:00:49.390
and actually, I'm going to
leave some blank space here.

00:00:49.390 --> 00:00:52.462
We don't have an x
squared term here.

00:00:52.462 --> 00:00:54.170
But I'm going to leave
some space for it,

00:00:54.170 --> 00:00:56.503
just so that we can align
everything in the proper place

00:00:56.503 --> 00:00:57.950
when we actually
do the division.

00:00:57.950 --> 00:01:00.470
So x to the third
plus nothing to the x

00:01:00.470 --> 00:01:05.140
squared power plus 5x minus 4.

00:01:05.140 --> 00:01:08.350
So we have a place for the
third power, the second power,

00:01:08.350 --> 00:01:10.360
the first power,
and the 0-th power.

00:01:10.360 --> 00:01:13.650
So now let's just do a little
bit of algebraic long division.

00:01:13.650 --> 00:01:15.720
Let's look at the
highest-degree term.

00:01:15.720 --> 00:01:18.910
x squared goes into x to
the third how many times?

00:01:18.910 --> 00:01:20.770
Well, it goes into it x times.

00:01:20.770 --> 00:01:24.300
x to the third
divided by x squared

00:01:24.300 --> 00:01:27.850
is equal to x to the 3 minus 2,
which is equal to x to the 1,

00:01:27.850 --> 00:01:30.200
which is equal to x.

00:01:30.200 --> 00:01:31.550
So it goes x times.

00:01:31.550 --> 00:01:33.660
I'll write the x
right over here.

00:01:33.660 --> 00:01:36.320
And we multiply x times
this entire thing.

00:01:36.320 --> 00:01:39.560
x times x squared
is x to the third.

00:01:39.560 --> 00:01:43.820
x times negative x is
negative x squared.

00:01:43.820 --> 00:01:48.390
x times 1 is positive x.

00:01:48.390 --> 00:01:50.660
And now we want to subtract
this whole expression

00:01:50.660 --> 00:01:51.880
from that whole expression.

00:01:51.880 --> 00:01:54.580
And that's the same thing
as adding the opposite,

00:01:54.580 --> 00:01:57.804
or multiplying each of
these terms by negative 1,

00:01:57.804 --> 00:01:59.220
and then adding
it to these terms.

00:01:59.220 --> 00:02:00.720
So let's do that.

00:02:00.720 --> 00:02:03.680
So we have negative
x to the third.

00:02:03.680 --> 00:02:07.400
Negative 1 times negative x
squared is positive x squared.

00:02:07.400 --> 00:02:10.560
And then positive x times
negative 1 is negative x.

00:02:10.560 --> 00:02:13.230
And so let's now add everything.

00:02:13.230 --> 00:02:16.600
x to the third minus x to
third-- those cancel out.

00:02:16.600 --> 00:02:19.700
0 plus x squared
gives us an x squared.

00:02:19.700 --> 00:02:23.070
5x minus x gives us a plus 4x.

00:02:23.070 --> 00:02:25.397
And then we bring
down this minus 4.

00:02:25.397 --> 00:02:26.980
We're not adding
anything to it there.

00:02:26.980 --> 00:02:28.530
You could view there's a 0 here.

00:02:28.530 --> 00:02:30.290
So let's bring down the minus 4.

00:02:30.290 --> 00:02:32.300
And now let's look at
the highest-degree terms.

00:02:32.300 --> 00:02:38.330
x squared goes into x
squared exactly one time.

00:02:38.330 --> 00:02:41.010
It's the same thing,
so we put a plus 1.

00:02:41.010 --> 00:02:44.800
And then you have 1 times
x squared is x squared.

00:02:44.800 --> 00:02:47.615
1 times negative
x is negative x.

00:02:47.615 --> 00:02:51.770
1 times 1 is 1.

00:02:51.770 --> 00:02:54.330
And now we want to
subtract this from that,

00:02:54.330 --> 00:02:56.080
or we want to add the opposite.

00:02:56.080 --> 00:02:58.330
And to add the
opposite, we can just

00:02:58.330 --> 00:03:00.640
multiply each of these
terms by negative 1.

00:03:00.640 --> 00:03:03.070
x squared becomes
negative x squared.

00:03:03.070 --> 00:03:05.770
Negative x times
negative 1 is positive x.

00:03:05.770 --> 00:03:09.840
And then positive 1 times
negative 1 is negative 1.

00:03:09.840 --> 00:03:11.620
Now let's do the
addition. x squared

00:03:11.620 --> 00:03:13.510
minus x squared--
they cancel out.

00:03:13.510 --> 00:03:17.420
4x plus x is 5x.

00:03:17.420 --> 00:03:24.490
And then we have negative
4 minus 1 is negative 5.

00:03:24.490 --> 00:03:27.230
Now, you might be
tempted to keep dividing,

00:03:27.230 --> 00:03:29.230
but you can't any more.

00:03:29.230 --> 00:03:31.670
This term right here, the
highest-degree term here,

00:03:31.670 --> 00:03:34.030
is now higher than the
highest-degree term

00:03:34.030 --> 00:03:36.530
that you're going to
try to divide into.

00:03:36.530 --> 00:03:37.950
So we have a remainder.

00:03:43.510 --> 00:03:48.000
So the answer to this is--
this expression right over here

00:03:48.000 --> 00:03:55.520
is equal to x plus 1
plus the remainder,

00:03:55.520 --> 00:04:00.770
plus 5x minus 5-- whatever
the remainder is-- divided

00:04:00.770 --> 00:04:07.100
by x squared minus x plus 1.

00:04:07.100 --> 00:04:09.130
If this was divisible,
we could keep dividing,

00:04:09.130 --> 00:04:10.213
but we're saying it's not.

00:04:10.213 --> 00:04:12.370
It's now a lower degree
than this down here.

00:04:12.370 --> 00:04:16.550
So we could say it's
x plus 1 plus whatever

00:04:16.550 --> 00:04:19.663
this remainder is, divided
by this thing over here.

00:04:19.663 --> 00:04:21.829
So our answer-- I'm going
to write it one more time.

00:04:21.829 --> 00:04:30.480
It's x plus 1 plus 5x minus 5,
over x squared minus x plus 1.

00:04:30.480 --> 00:04:31.920
And we can check
that this works.

00:04:31.920 --> 00:04:35.000
If we take this thing over
here, and we multiply it

00:04:35.000 --> 00:04:37.600
by this thing over
here, we should

00:04:37.600 --> 00:04:40.390
get the x to the
third plus 5x minus 4.

00:04:40.390 --> 00:04:41.610
So let's do that.

00:04:41.610 --> 00:04:50.170
Let's multiply this thing
by x squared minus x plus 1.

00:04:50.170 --> 00:04:51.910
And to do that,
let's just distribute

00:04:51.910 --> 00:04:55.935
this whole trinomial
times each of these terms.

00:04:58.980 --> 00:05:02.300
When we do the
first term, we have

00:05:02.300 --> 00:05:05.030
x squared minus
x plus 1 times x.

00:05:05.030 --> 00:05:06.840
So that's going to
be x times x squared,

00:05:06.840 --> 00:05:09.500
which is x to the third;
x times negative x, which

00:05:09.500 --> 00:05:13.700
is negative x squared; x
times 1, which is plus x.

00:05:13.700 --> 00:05:16.420
Then we can multiply
this whole thing times 1.

00:05:16.420 --> 00:05:20.360
So it's going to be plus
x squared minus x plus 1.

00:05:20.360 --> 00:05:22.590
I'm just multiplying
all of these times 1.

00:05:22.590 --> 00:05:26.580
And then we can multiply this
whole thing times this thing.

00:05:26.580 --> 00:05:28.860
Now, this is the same
as the denominator here.

00:05:28.860 --> 00:05:29.890
So it'll cancel out.

00:05:29.890 --> 00:05:31.100
This will cancel with that.

00:05:31.100 --> 00:05:36.440
And we're just going to be left
with the numerator over here,

00:05:36.440 --> 00:05:40.170
so plus 5x minus 5.

00:05:40.170 --> 00:05:42.330
And now we can try
to simplify it.

00:05:42.330 --> 00:05:45.600
We only have one third-degree
term, the x to the third.

00:05:45.600 --> 00:05:47.360
So we have x to the third here.

00:05:47.360 --> 00:05:50.809
Second-degree terms-- we
have a negative x squared.

00:05:50.809 --> 00:05:52.600
And then we also have
a positive x squared.

00:05:52.600 --> 00:05:54.850
So they cancel out
with each other.

00:05:54.850 --> 00:05:56.450
First-degree terms-- let's see.

00:05:56.450 --> 00:05:59.230
We have a positive
x and a negative x.

00:05:59.230 --> 00:06:01.290
Those cancel out
with each other.

00:06:01.290 --> 00:06:03.600
So we're just going to
have that 5x over here.

00:06:03.600 --> 00:06:05.440
So we're just going
to have this 5x.

00:06:05.440 --> 00:06:08.160
So then we have plus 5x.

00:06:08.160 --> 00:06:11.690
And then we have the 0-th degree
terms, or the constant terms.

00:06:11.690 --> 00:06:14.150
We have a positive
1 and a negative 5.

00:06:14.150 --> 00:06:14.900
Add them together.

00:06:14.900 --> 00:06:16.570
You get negative 4.

00:06:16.570 --> 00:06:19.520
So you get x to the
third plus 5x minus 4,

00:06:19.520 --> 00:06:22.970
which is exactly what
we had over here.


Example 6: Subtracting polynomials with two variables

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:05.080
We've got 4x squared
y minus 3xy plus 25

00:00:05.080 --> 00:00:12.470
minus the entire expression 9y
squared x plus 7xy minus 20.

00:00:12.470 --> 00:00:14.910
So when we're subtracting
this entire expression,

00:00:14.910 --> 00:00:17.502
that's equivalent to
subtracting each of these terms

00:00:17.502 --> 00:00:19.460
individually if we didn't
have the parentheses.

00:00:19.460 --> 00:00:21.250
Or another way of
thinking about it--

00:00:21.250 --> 00:00:23.240
we could distribute
this negative sign.

00:00:23.240 --> 00:00:26.460
Or you could view this
as a negative 1 times

00:00:26.460 --> 00:00:27.660
this entire expression.

00:00:27.660 --> 00:00:28.770
And we can distribute it.

00:00:28.770 --> 00:00:29.760
So let's do that.

00:00:29.760 --> 00:00:31.620
So let me write this
first expression here.

00:00:31.620 --> 00:00:33.020
I'm going to write it unchanged.

00:00:33.020 --> 00:00:39.310
So it is 4x squared
y minus 3xy plus 25.

00:00:39.310 --> 00:00:41.190
And now let me distribute
the negative 1,

00:00:41.190 --> 00:00:44.120
or the negative sign
times all of this stuff.

00:00:44.120 --> 00:00:51.460
So negative 1 times 9y squared
x is negative 9y squared x.

00:00:51.460 --> 00:00:57.720
Negative 1 times
7xy is negative 7xy.

00:00:57.720 --> 00:01:02.220
And then negative 1
times 20 is positive 20.

00:01:02.220 --> 00:01:04.110
And now we just have
to add these terms.

00:01:04.110 --> 00:01:06.120
And we just want to
group like terms.

00:01:06.120 --> 00:01:08.840
So let's see, is there another
x squared y term anywhere?

00:01:08.840 --> 00:01:10.240
No, I don't see one.

00:01:10.240 --> 00:01:13.270
So I'll just rewrite this.

00:01:13.270 --> 00:01:16.580
So we have 4x squared y.

00:01:16.580 --> 00:01:18.370
Now, is there another xy term?

00:01:18.370 --> 00:01:19.260
Yeah, there is.

00:01:19.260 --> 00:01:25.720
So we can group negative
3xy and negative 7xy.

00:01:25.720 --> 00:01:29.230
Negative 3 of something minus
another 7 of that something

00:01:29.230 --> 00:01:31.170
is going to be negative
10 of that something.

00:01:31.170 --> 00:01:34.730
So it's negative 10xy.

00:01:34.730 --> 00:01:39.020
And then we have a 25, which
is just a constant term.

00:01:39.020 --> 00:01:40.430
Or an x to the 0 term.

00:01:40.430 --> 00:01:41.306
It's 25x to the 0.

00:01:41.306 --> 00:01:42.430
You could view it that way.

00:01:42.430 --> 00:01:44.513
And there's another constant
term right over here.

00:01:44.513 --> 00:01:47.170
We can always add 25 to 20.

00:01:47.170 --> 00:01:49.540
That gives us 45.

00:01:49.540 --> 00:01:51.519
And then we have this
term right over here,

00:01:51.519 --> 00:01:53.560
which clearly can't be
merged with anything else.

00:01:53.560 --> 00:01:57.050
So minus 9y squared.

00:01:57.050 --> 00:01:58.780
Let me do that in
that original color.

00:01:58.780 --> 00:02:02.540
Minus 9-- I'm having
trouble shifting

00:02:02.540 --> 00:02:07.210
colors-- minus 9y squared x.

00:02:07.210 --> 00:02:09.510
And we are done.


Dividing multivariable polynomial with monomial

WEBVTT
Kind: captions
Language: en

00:00:00.867 --> 00:00:07.105
We've got... All of that divided by

00:00:07.105 --> 00:00:09.400
4xy

00:00:09.400 --> 00:00:12.533
There are several ways to think about them, they're all equivalent

00:00:12.533 --> 00:00:15.400
This is going to be the exact same thing. We can esssentialy sort of separate the numerators.

00:00:15.400 --> 00:00:34.667
This is the same thing as...

00:00:34.667 --> 00:00:36.333
And there's multiple ways to think about it.

00:00:36.333 --> 00:00:40.000
One is: If I have a+b over c

00:00:40.000 --> 00:00:43.800
this is clearly the same thing as a over c plus b over c

00:00:43.800 --> 00:00:46.933
we're just adding the--we're taking the fraction apart.

00:00:46.933 --> 00:00:49.691
If we did it in reverse, we'd just be adding fractions

00:00:49.691 --> 00:00:51.400
Another way to think about it is

00:00:51.400 --> 00:00:52.600
We're distributing the division.

00:00:52.600 --> 00:00:56.000
We're dividing the whole thing by 4xy

00:00:56.000 --> 00:00:58.096
so we divide each of the terms by 4xy.

00:00:58.096 --> 00:01:00.349
The last way to think about it, which is really the same thing,

00:01:00.349 --> 00:01:04.087
is that we're multiplying the whole thing by 1 over 4xy

00:01:04.087 --> 00:01:07.600
So we're distributing the 1 over 4xy times everything

00:01:07.600 --> 00:01:10.800
which is the same as dividing by 4xy.

00:01:10.800 --> 00:01:13.667
Either way we get to this and we just have to simplify this

00:01:13.667 --> 00:01:15.867
really just using exponent properties.

00:01:15.867 --> 00:01:20.867
So we can divide 12. 12 divided by 4 is pretty straight forward

00:01:20.867 --> 00:01:25.267
that is just 3, so let me just write it this way.

00:01:25.267 --> 00:01:30.400
So 12 divided by 4 is 3. And x squared

00:01:30.400 --> 00:01:35.434
divided by x. Remember x is the same thing as x to the first power,

00:01:35.434 --> 00:01:39.846
so thats going to be x to the three minus two

00:01:39.846 --> 00:01:45.200
Sorry, x to the three minus one power.

00:01:45.200 --> 00:01:52.013
Which would be x squared. And finally

00:01:52.013 --> 00:01:54.533
you have y to the third divided by y.

00:01:54.533 --> 00:01:57.098
Well thats just divided by y to the one.

00:01:57.098 --> 00:02:01.022
and y to the third divided by y to the one is going to be y squared.

00:02:01.022 --> 00:02:03.267
Three minus one is two.

00:02:03.267 --> 00:02:05.733
Now lets do the next term.

00:02:05.733 --> 00:02:13.267
Plus, we have sixteen over 4, which is just four.

00:02:13.267 --> 00:02:18.925
That is just equal to four. Then we have x to the fourth over

00:02:18.925 --> 00:02:23.733
x. Well, once again, x is just the same thing as x to the first

00:02:23.733 --> 00:02:28.840
So this is going to be x to the four minus one or x to the third power.

00:02:28.840 --> 00:02:33.133
And then finally, we have y divided by y.

00:02:33.133 --> 00:02:34.733
Well, thats just going to be one.

00:02:34.733 --> 00:02:39.133
So we don't even have to write it down. We could right one here,

00:02:39.133 --> 00:02:40.682
but that wont do much, won't change the values.

00:02:40.682 --> 00:02:43.400
So just these guys essentially cancel out. And then finally you have

00:02:43.400 --> 00:02:48.800
ten divided by 4. they're both divisible by two, so it becomes five halves.

00:02:48.800 --> 00:02:53.523
So that part over there simplifies to five over two.

00:02:53.523 --> 00:02:58.533
So, plus five halves and then x divided by x.

00:02:58.533 --> 00:03:01.867
Those just cancel out, so thats just one.

00:03:01.867 --> 00:03:03.533
y divided by y, thats just one as well.

00:03:03.533 --> 00:03:05.133
So we're left with five halves.

00:03:05.133 --> 00:03:06.015
And we're done!

00:03:06.015 --> 99:59:59.999
This whole expression simplified to...


Example 2: Multiplying a binomial by a polynomial

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:05.120
We are multiplying 10a minus
3 by the entire polynomial 5a

00:00:05.120 --> 00:00:07.567
squared plus 7a minus 1.

00:00:07.567 --> 00:00:09.900
So to do this, we can just
do the distributive property.

00:00:09.900 --> 00:00:12.140
We can distribute this
entire polynomial,

00:00:12.140 --> 00:00:15.020
this entire trinomial,
times each of these terms.

00:00:15.020 --> 00:00:21.930
We could have 5a squared
plus 7a minus 1 times 10a.

00:00:21.930 --> 00:00:26.210
And then 5a squared plus 7a
minus 1 times negative 3.

00:00:26.210 --> 00:00:27.890
So let's just do that.

00:00:27.890 --> 00:00:31.319
So if we have-- so let
me just write it out.

00:00:31.319 --> 00:00:32.360
Let me write it this way.

00:00:32.360 --> 00:00:43.900
10a times 5a squared
plus 7a minus 1.

00:00:43.900 --> 00:00:46.420
That's that right over here.

00:00:46.420 --> 00:00:52.180
And then we can have
minus 3 times 5a squared

00:00:52.180 --> 00:00:54.710
plus 7a minus 1.

00:00:54.710 --> 00:00:57.190
And that is this
distribution right over here.

00:00:57.190 --> 00:00:58.800
And then we can simplify it.

00:00:58.800 --> 00:01:03.410
10a times 5a squared--
10 times 5 is 50.

00:01:03.410 --> 00:01:06.360
a times a squared
is a to the third.

00:01:06.360 --> 00:01:09.820
10 times 7 is 70.

00:01:09.820 --> 00:01:12.160
a times a is a squared.

00:01:12.160 --> 00:01:16.330
10a times negative
1 is negative 10a.

00:01:16.330 --> 00:01:19.070
Then we distribute this
negative 3 times all of this.

00:01:19.070 --> 00:01:24.900
Negative 3 times 5a squared
is negative 15a squared.

00:01:24.900 --> 00:01:29.280
Negative 3 times
7a is negative 21a.

00:01:29.280 --> 00:01:33.380
Negative 3 times
negative 1 is positive 3.

00:01:33.380 --> 00:01:35.950
And now we can try
to merge like terms.

00:01:35.950 --> 00:01:38.330
This is the only a to
the third term here.

00:01:38.330 --> 00:01:39.630
So this is 50a to the third.

00:01:39.630 --> 00:01:41.340
I'll just rewrite it.

00:01:41.340 --> 00:01:43.660
Now we have two a squared terms.

00:01:43.660 --> 00:01:47.810
We have 70a squared minus
15, or negative 15a squared.

00:01:47.810 --> 00:01:49.880
So we can add these two terms.

00:01:49.880 --> 00:01:53.190
70 of something minus
15 of that something

00:01:53.190 --> 00:01:56.950
is going to be 55
of that something.

00:01:56.950 --> 00:02:01.490
So plus 55a squared.

00:02:01.490 --> 00:02:03.800
And then we also
have two a terms.

00:02:03.800 --> 00:02:07.520
We have this negative 10a, and
then we have this negative 21a.

00:02:07.520 --> 00:02:12.430
So if we go negative 10 minus
21, that is negative 31.

00:02:12.430 --> 00:02:14.170
That is negative 31a.

00:02:23.670 --> 00:02:27.290
And then finally, we only have
one constant term over here.

00:02:27.290 --> 00:02:29.520
We have this positive 3.

00:02:29.520 --> 00:02:31.610
So plus 3.

00:02:31.610 --> 00:02:34.270
And we are done.


Evaluating exponential expressions 3

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:03.060
Rewrite the following
expression as the product

00:00:03.060 --> 00:00:06.360
of positive exponents, and
then evaluate the expression

00:00:06.360 --> 00:00:08.410
when x is equal to 2.

00:00:08.410 --> 00:00:09.810
So let's do this first part.

00:00:09.810 --> 00:00:12.872
They want to rewrite it as a
product of positive exponents.

00:00:12.872 --> 00:00:14.330
So they don't like
the fact that we

00:00:14.330 --> 00:00:16.890
have this negative
third power and the 5

00:00:16.890 --> 00:00:18.300
to the negative second power.

00:00:18.300 --> 00:00:20.570
So we want to rewrite those
as negative exponents.

00:00:20.570 --> 00:00:23.920
And to do that, we just have
to remember that something

00:00:23.920 --> 00:00:27.370
to a negative power is just--
let me write it over here.

00:00:30.210 --> 00:00:33.050
If I have x to the
negative two power,

00:00:33.050 --> 00:00:35.630
this is the exact
same thing as 1

00:00:35.630 --> 00:00:39.120
over x to the
positive two power.

00:00:39.120 --> 00:00:43.470
If I have a to the
negative b power,

00:00:43.470 --> 00:00:48.164
this is the exact same thing
as 1 over a to the b power.

00:00:48.164 --> 00:00:50.330
You can get rid of that
negative by just putting a 1

00:00:50.330 --> 00:00:51.246
over this whole thing.

00:00:51.246 --> 00:00:53.380
By taking the inverse
of the whole expression,

00:00:53.380 --> 00:00:54.760
you can get rid
of that negative.

00:00:54.760 --> 00:00:58.680
So let's do that over here. x to
the negative three power, that

00:00:58.680 --> 00:01:01.750
is equal to this-- let me
do this in different colors.

00:01:01.750 --> 00:01:03.430
x to the negative
third power, that's

00:01:03.430 --> 00:01:07.440
the exact same thing as 1
over x to the third power

00:01:07.440 --> 00:01:08.700
by definition.

00:01:08.700 --> 00:01:13.470
And we're going to multiply that
by 5 to the negative two power.

00:01:13.470 --> 00:01:15.230
Well, 5 to the
negative two power.

00:01:15.230 --> 00:01:16.330
Same exact thing.

00:01:16.330 --> 00:01:17.740
That's the same.

00:01:17.740 --> 00:01:21.240
That's the same thing as 1
over 5 to the second power,

00:01:21.240 --> 00:01:25.150
or 1 over 5 to the
positive two power.

00:01:25.150 --> 00:01:28.520
And then finally, we have
just a straight up x squared.

00:01:28.520 --> 00:01:30.370
The exponent is
already positive there.

00:01:30.370 --> 00:01:34.280
So, we could just leave
that as times x squared.

00:01:34.280 --> 00:01:35.340
So we did the first part.

00:01:35.340 --> 00:01:36.798
We rewrote the
following expression

00:01:36.798 --> 00:01:39.480
as the product of
positive exponents.

00:01:39.480 --> 00:01:41.070
Now let's evaluate
the expression

00:01:41.070 --> 00:01:43.340
when x is equal to 2.

00:01:43.340 --> 00:01:45.430
So everywhere we see an
x, we should put a 2.

00:01:45.430 --> 00:01:48.960
So this becomes 1
over 2 to the-- let

00:01:48.960 --> 00:01:51.200
me do the same colors.

00:01:51.200 --> 00:01:55.450
This becomes 1 over
2 to the third power

00:01:55.450 --> 00:02:04.550
times 1 over 5 squared
times 2 squared.

00:02:04.550 --> 00:02:07.180
Every place we saw an x,
we're now putting a 2.

00:02:07.180 --> 00:02:10.590
2 to the third power, just
a reminder, that's 2 times

00:02:10.590 --> 00:02:12.620
2 times 2.

00:02:12.620 --> 00:02:15.190
That is equal to 8.

00:02:15.190 --> 00:02:20.990
So this is equal to 1 over
8 times 1 over 5 squared.

00:02:20.990 --> 00:02:26.980
5 squared is equal to 5 times
5, which is equal to 25.

00:02:26.980 --> 00:02:28.980
So this is equal to 25.

00:02:28.980 --> 00:02:32.070
And then we're going to
multiply that times 2 squared.

00:02:32.070 --> 00:02:41.830
So 2 squared is 2 times 2,
which is equal to 4, so times 4.

00:02:41.830 --> 00:02:44.886
And you could even do
this as times 4 over 1.

00:02:44.886 --> 00:02:46.510
And so you multiply
all the numerators.

00:02:46.510 --> 00:02:48.470
That's what we do when
we multiply fractions.

00:02:48.470 --> 00:02:50.510
You multiply all the
numerators, 1 times 1 times

00:02:50.510 --> 00:02:53.300
4 is equal to 4.

00:02:53.300 --> 00:02:54.900
And you multiply
the denominators.

00:02:54.900 --> 00:03:01.970
8 times 25 is going to be 200.

00:03:01.970 --> 00:03:03.326
And then you can simplify this.

00:03:03.326 --> 00:03:04.700
The numerator and
the denominator

00:03:04.700 --> 00:03:06.510
are both divisible by 4.

00:03:06.510 --> 00:03:08.870
Divide the numerator
by 4, you get one.

00:03:08.870 --> 00:03:11.700
Divide the denominator
by 4, you get 50.

00:03:11.700 --> 00:03:14.589
So this is equal to 1 over 50.

00:03:14.589 --> 00:03:16.130
And just so you see,
there's actually

00:03:16.130 --> 00:03:18.730
a way to simplify it before
we even got to this stage

00:03:18.730 --> 00:03:19.310
over here.

00:03:19.310 --> 00:03:22.660
Actually from this stage, we
could have multiplied it out.

00:03:22.660 --> 00:03:24.780
Let me just rewrite this thing.

00:03:24.780 --> 00:03:27.690
So this product, if we multiply
it out, this 2 squared,

00:03:27.690 --> 00:03:30.520
you could view it
as 2 squared over 1.

00:03:30.520 --> 00:03:32.580
So if we were to
multiply this out,

00:03:32.580 --> 00:03:35.010
the numerator would
have been equal

00:03:35.010 --> 00:03:38.250
to 1 times 1 times 2 squared.

00:03:38.250 --> 00:03:40.010
2 squared is just 2 times 2.

00:03:40.010 --> 00:03:42.570
So it would've been 2
times 2 in the numerator.

00:03:42.570 --> 00:03:45.330
We can ignore the 1's, that
doesn't change the value.

00:03:45.330 --> 00:03:48.610
And then the denominator would
have been 2 times 2 times 2.

00:03:48.610 --> 00:03:50.110
That's what 2 to
the third power is.

00:03:50.110 --> 00:03:54.720
2 times 2 times 2
times 5 times 5.

00:03:57.930 --> 00:03:59.450
That's what 5 squared is.

00:03:59.450 --> 00:04:02.334
And of course multiplying it
by 1 doesn't change this value.

00:04:02.334 --> 00:04:04.000
And then it would
have been more obvious

00:04:04.000 --> 00:04:07.230
that you could divide the
numerator and denominator by 2

00:04:07.230 --> 00:04:09.760
once, you could do
it again, and then

00:04:09.760 --> 00:04:16.464
you'd just be left with
1 over 2 times 5 times 5.

00:04:19.430 --> 00:04:21.339
And this, of course,
down here is 50.

00:04:21.339 --> 00:04:23.840
This is 10 times 5, which is 50.

00:04:23.840 --> 00:04:26.250
You would have gotten
the same result.


Evaluating exponential expressions 2

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:04.640
Evaluate the expression 4n
to the first power minus 2n

00:00:04.640 --> 00:00:08.900
to the zeroth power for n
equals 1 and n equals 5.

00:00:08.900 --> 00:00:10.630
So, let's do n equals 1 first.

00:00:10.630 --> 00:00:13.660
So, every place we see an
n, let's put a 1 in there.

00:00:13.660 --> 00:00:21.270
So this is equal to 4 times 1 to
the first power minus 2 times 1

00:00:21.270 --> 00:00:22.630
to the zeroth power.

00:00:22.630 --> 00:00:24.480
Every place we saw an
n, we have an n there.

00:00:24.480 --> 00:00:28.240
We have an n to the
first, one to the first.

00:00:28.240 --> 00:00:31.100
So remember, we want to do
our order of operations.

00:00:31.100 --> 00:00:31.900
We have PEMDAS.

00:00:31.900 --> 00:00:34.233
And I'm going to write it
this way just to make it clear

00:00:34.233 --> 00:00:36.600
that we do parentheses
then exponentiation.

00:00:36.600 --> 00:00:39.870
Then multiplication and
division are at the same level,

00:00:39.870 --> 00:00:41.300
they get the same priority.

00:00:41.300 --> 00:00:43.670
And then addition
and subtraction

00:00:43.670 --> 00:00:45.170
get the same priority.

00:00:45.170 --> 00:00:48.060
So, we want to do our exponents
before we do anything else.

00:00:48.060 --> 00:00:49.640
There are no parentheses here.

00:00:49.640 --> 00:00:52.510
So, we first want to evaluate
one to the first power.

00:00:52.510 --> 00:00:55.230
And anything to the first
power is just that anything.

00:00:55.230 --> 00:00:57.970
If I just have any number
to the first power,

00:00:57.970 --> 00:01:00.340
that is just going to
be equal to that number.

00:01:00.340 --> 00:01:03.720
So, 1 to the first power
is just equal to 1.

00:01:03.720 --> 00:01:05.724
So, this is going
to be 4 times 1.

00:01:05.724 --> 00:01:06.390
Let me write it.

00:01:06.390 --> 00:01:13.930
4 times 1 minus 2 times
1 to the zeroth power.

00:01:13.930 --> 00:01:17.610
Now, any number other
than 0 to the zeroth power

00:01:17.610 --> 00:01:19.230
is going to be 1.

00:01:19.230 --> 00:01:23.870
So, if I tell you x to the
zeroth power is equal to what?

00:01:23.870 --> 00:01:26.850
And I tell you that
x is not equal to 0.

00:01:26.850 --> 00:01:28.870
What is x to the zeroth power?

00:01:28.870 --> 00:01:31.390
You should immediately
know that it is equal to 1.

00:01:31.390 --> 00:01:34.330
I exclude 0 because
0 to the zeroth power

00:01:34.330 --> 00:01:36.240
is usually undefined.

00:01:36.240 --> 00:01:38.659
Or for our purposes, you can
accept it to be undefined.

00:01:38.659 --> 00:01:40.325
And maybe in the
future, I'll do a video

00:01:40.325 --> 00:01:43.240
on explaining why
it's undefined.

00:01:43.240 --> 00:01:45.920
There's some good arguments
for why this should be 0.

00:01:45.920 --> 00:01:48.675
Because 0 to pretty much
any other power is a 0.

00:01:48.675 --> 00:01:50.300
But then there's a
really good argument

00:01:50.300 --> 00:01:51.258
for why it should be 1.

00:01:51.258 --> 00:01:54.830
Because any other number
to the zeroth power is 1.

00:01:54.830 --> 00:01:56.520
So that's why,
for the most part,

00:01:56.520 --> 00:01:59.120
we like to keep this undefined.

00:01:59.120 --> 00:02:01.000
So with that said, 1
to the zeroth power

00:02:01.000 --> 00:02:03.630
is just going to be 1.

00:02:03.630 --> 00:02:06.460
Anything other than 0 to
the zeroth power is 1.

00:02:06.460 --> 00:02:11.250
So, we have 4 times 1, which is
4, minus 2 times 1, which is 2.

00:02:11.250 --> 00:02:14.700
So, 4 minus 2 is
just equal to 2.

00:02:14.700 --> 00:02:18.100
Now, let's do the same
thing with n equals 5.

00:02:18.100 --> 00:02:24.650
So it becomes 4 times 5 to the
first power minus 2 times 5

00:02:24.650 --> 00:02:26.480
to the zeroth power.

00:02:26.480 --> 00:02:27.010
Same thing.

00:02:27.010 --> 00:02:31.890
5 to the first power is
just going to be equal to 5.

00:02:31.890 --> 00:02:35.780
So, I want to do that
in a different color.

00:02:35.780 --> 00:02:39.100
5 to the first power is
just going to be equal to 5.

00:02:39.100 --> 00:02:42.550
5 to the zeroth power is
just going to be equal to 1.

00:02:42.550 --> 00:02:48.440
And so, this expression becomes
4 times 5 minus 2 times 1.

00:02:48.440 --> 00:02:55.470
Or 4 times 5 is 20, minus
2 times 1, that is 2.

00:02:55.470 --> 00:02:59.670
And this is equal to 18.

00:02:59.670 --> 00:03:01.460
And we're done.


Exponent properties 2

WEBVTT
Kind: captions
Language: en

00:00:00.740 --> 00:00:04.660
Simplify 3a to the fifth
over 9a squared times a

00:00:04.660 --> 00:00:07.192
to the fourth over
a to the third.

00:00:07.192 --> 00:00:08.900
So before we even
worry about the a's, we

00:00:08.900 --> 00:00:11.270
can actually simplify
the 3 and the 9.

00:00:11.270 --> 00:00:13.030
They're both divisible by 3.

00:00:13.030 --> 00:00:17.650
So let's divide the numerator
and the denominator here by 3.

00:00:17.650 --> 00:00:20.880
So if we divide the numerator
by 3, the 3 becomes a 1.

00:00:20.880 --> 00:00:25.920
If we divide the denominator
by 3, the 9 becomes a 3.

00:00:25.920 --> 00:00:28.270
So this reduces
to, or simplifies

00:00:28.270 --> 00:00:30.670
to 1a to the fifth
times a to the fourth

00:00:30.670 --> 00:00:33.580
over-- or maybe I should
say, a to the fifth over 3a

00:00:33.580 --> 00:00:37.130
squared times a to the
fourth over a to the third.

00:00:37.130 --> 00:00:40.370
Now this, if we just
multiply the two expressions,

00:00:40.370 --> 00:00:43.677
this would be equal to 1a to
the fifth times a to the fourth

00:00:43.677 --> 00:00:46.135
in the numerator, and we don't
have to worry about the one,

00:00:46.135 --> 00:00:47.340
it doesn't change the value.

00:00:47.340 --> 00:00:52.200
So it's a to the fifth times a
to the fourth in the numerator.

00:00:52.200 --> 00:00:56.230
And then we have 3a-- let
me write the 3 like this--

00:00:56.230 --> 00:01:00.270
and then we have 3
times a squared times

00:01:00.270 --> 00:01:03.330
a to the third in
the denominator.

00:01:03.330 --> 00:01:05.349
And now there's
multiple ways that we

00:01:05.349 --> 00:01:07.650
can simplify this from here.

00:01:07.650 --> 00:01:10.190
One sometimes is called
the quotient rule.

00:01:10.190 --> 00:01:16.590
And that's just the idea that
if you have a to the x over a

00:01:16.590 --> 00:01:22.470
to the y, that this is going to
be equal to a to the x minus y.

00:01:22.470 --> 00:01:25.510
And just to understand
why that works,

00:01:25.510 --> 00:01:27.980
let's think about a to
the fifth over a squared.

00:01:27.980 --> 00:01:33.600
So a to the fifth
is literally a times

00:01:33.600 --> 00:01:37.176
a times a times a times a.

00:01:37.176 --> 00:01:39.870
That right there
is a to the fifth.

00:01:39.870 --> 00:01:43.270
And we have that over a squared.

00:01:43.270 --> 00:01:46.790
And I'm just thinking about
the a squared right over here,

00:01:46.790 --> 00:01:49.556
which is literally
just a times a.

00:01:49.556 --> 00:01:51.600
That is a squared.

00:01:51.600 --> 00:01:55.110
Now, clearly, both the
numerator and denominator

00:01:55.110 --> 00:01:57.420
are both divisible by a times a.

00:01:57.420 --> 00:01:59.710
We can divide them
both by a times a.

00:01:59.710 --> 00:02:02.690
So we can get rid of-- if
we divide the numerator

00:02:02.690 --> 00:02:06.870
by a twice, by a times a, so
let's get rid of a times a.

00:02:06.870 --> 00:02:10.400
And if we divide the denominator
by a times a, we just get a 1.

00:02:10.400 --> 00:02:11.950
So what are we just left with?

00:02:11.950 --> 00:02:16.990
We are left with just a
times a times a over 1,

00:02:16.990 --> 00:02:20.670
which is just a times a times a.

00:02:20.670 --> 00:02:21.980
But what is this?

00:02:21.980 --> 00:02:29.220
This is a to the third power,
or a to the 5 minus 2 power.

00:02:29.220 --> 00:02:33.700
We had 5, we were able to
cancel out 2, that gave us 3.

00:02:33.700 --> 00:02:35.430
So we could do the
same thing over here.

00:02:35.430 --> 00:02:36.620
We can apply the quotient rule.

00:02:36.620 --> 00:02:38.450
And I'll do two ways
of actually doing this.

00:02:38.450 --> 00:02:40.241
So let's apply the
quotient rule with the a

00:02:40.241 --> 00:02:42.640
to the fifth and the a squared.

00:02:42.640 --> 00:02:44.830
So let me do it this way.

00:02:44.830 --> 00:02:47.160
So let's apply with
these two guys,

00:02:47.160 --> 00:02:51.480
and then let's apply
it with these two guys.

00:02:51.480 --> 00:02:56.100
And of course, we have
the 1/3 out front.

00:02:56.100 --> 00:03:00.180
So this can be
reduced to 1/3 times--

00:03:00.180 --> 00:03:01.960
if we apply the
quotient rule with a

00:03:01.960 --> 00:03:04.080
to the fifth over
a squared, we just

00:03:04.080 --> 00:03:07.580
did it over here-- that
becomes a to the third power.

00:03:07.580 --> 00:03:11.650
And if we apply it over here
with the a to the fourth over a

00:03:11.650 --> 00:03:15.150
to the third, that'll
give us a-- let

00:03:15.150 --> 00:03:16.930
me do it that same blue color.

00:03:16.930 --> 00:03:21.660
That'll give us a-- that's
not the same blue color.

00:03:21.660 --> 00:03:22.410
There we go.

00:03:22.410 --> 00:03:25.710
This will give us a to
the 4 minus 3 power,

00:03:25.710 --> 00:03:28.120
or a to the first power.

00:03:28.120 --> 00:03:31.360
And of course, we can
simplify this as a

00:03:31.360 --> 00:03:34.470
to the third times
a-- well, actually,

00:03:34.470 --> 00:03:36.250
let me just do it over here.

00:03:36.250 --> 00:03:39.070
Before I even rewrite it, we
know that a to the third times

00:03:39.070 --> 00:03:42.260
a to the first is going to
be a to the 3 plus 1 power.

00:03:42.260 --> 00:03:45.040
We have the same base,
we can add the exponents.

00:03:45.040 --> 00:03:47.440
We're multiplying a times
itself three times and then

00:03:47.440 --> 00:03:48.300
one more time.

00:03:48.300 --> 00:03:50.300
So that'll be a to
the fourth power.

00:03:50.300 --> 00:03:53.900
So this right over here
becomes a to the fourth power.

00:03:53.900 --> 00:03:55.940
a to the 3 plus 1 power.

00:03:55.940 --> 00:03:58.710
And then we have to
multiply that by 1/3.

00:03:58.710 --> 00:04:00.910
So our answer could be
1/3 a to the fourth,

00:04:00.910 --> 00:04:06.820
or we could equally write
it a to the fourth over 3.

00:04:06.820 --> 00:04:08.730
Now, the other way
to do this problem

00:04:08.730 --> 00:04:10.500
would have been to
apply the product,

00:04:10.500 --> 00:04:12.510
or to add the exponents
in the numerator,

00:04:12.510 --> 00:04:14.491
and then add the exponents
in the denominator.

00:04:14.491 --> 00:04:15.740
So let's do it that way first.

00:04:15.740 --> 00:04:17.690
If we add the exponents
in the numerator first,

00:04:17.690 --> 00:04:19.314
we don't apply the
quotient rule first.

00:04:19.314 --> 00:04:20.190
We apply it second.

00:04:20.190 --> 00:04:24.060
We get in the numerator, a to
the fifth times a to the fourth

00:04:24.060 --> 00:04:26.760
would be a to the ninth power.

00:04:26.760 --> 00:04:27.940
5 plus 4.

00:04:27.940 --> 00:04:32.680
And then in the denominator
we have a squared times a

00:04:32.680 --> 00:04:33.326
to the third.

00:04:33.326 --> 00:04:35.450
Add the exponents, because
we're taking the product

00:04:35.450 --> 00:04:36.640
with the same base.

00:04:36.640 --> 00:04:38.374
So it'll be a to
the fifth power.

00:04:38.374 --> 00:04:40.290
And of course, we still
have this 3 down here.

00:04:40.290 --> 00:04:44.860
We have a 1/3, or we could
just write a 3 over here.

00:04:44.860 --> 00:04:48.060
Now, we could apply the
quotient property of exponents.

00:04:48.060 --> 00:04:52.110
We could say, look, we have a to
the ninth over a to the fifth.

00:04:52.110 --> 00:04:53.980
a to the ninth
over a to the fifth

00:04:53.980 --> 00:04:57.530
is equal to a to
the 9 minus 5 power,

00:04:57.530 --> 00:05:00.960
or it's equal to a
to the fourth power.

00:05:00.960 --> 00:05:03.830
And of course, we still
have the divided by 3.

00:05:07.860 --> 00:05:11.230
Either way we got
the same answer.


Evaluating an algebraic expression with exponents

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:03.920
Evaluate the expression
5y to the fourth minus y

00:00:03.920 --> 00:00:07.320
squared when y is equal to 3.

00:00:07.320 --> 00:00:08.830
So every place we
see a y here, we

00:00:08.830 --> 00:00:11.990
could just replace it
with a 3 to evaluate it.

00:00:11.990 --> 00:00:19.080
So it becomes 5 times 3 to the
fourth power minus 3 squared.

00:00:19.080 --> 00:00:22.470
All I did is every time we
saw a y here, I put a 3 there.

00:00:22.470 --> 00:00:24.770
Every time we saw
a y, I put a 3.

00:00:24.770 --> 00:00:26.030
So what does this evaluate to?

00:00:26.030 --> 00:00:28.170
And we have to remember
our order of operations.

00:00:28.170 --> 00:00:29.780
Remember, parentheses
comes first.

00:00:29.780 --> 00:00:31.510
Sometimes it's
referred to as PEMDAS.

00:00:31.510 --> 00:00:32.930
Let me write that down.

00:00:32.930 --> 00:00:35.850
PEMDAS, PEMDAS.

00:00:35.850 --> 00:00:37.400
P is for parentheses.

00:00:37.400 --> 00:00:39.150
E is for exponents.

00:00:39.150 --> 00:00:41.880
M and D are for
Multiplication and Division.

00:00:41.880 --> 00:00:43.800
They're really at the
same level of priority.

00:00:43.800 --> 00:00:45.000
And then addition
and subtraction

00:00:45.000 --> 00:00:45.916
are at the same level.

00:00:45.916 --> 00:00:48.820
If you really want to do it
properly, it should be P-E,

00:00:48.820 --> 00:00:50.320
and then multiplication
and division

00:00:50.320 --> 00:00:51.528
are really at the same level.

00:00:51.528 --> 00:00:54.440
And addition and subtraction
are at the same level.

00:00:54.440 --> 00:00:56.960
But what this tells us is
that we do parentheses first.

00:00:56.960 --> 00:00:58.720
But then after
that, exponentiation

00:00:58.720 --> 00:01:01.180
takes priority over
everything else here.

00:01:01.180 --> 00:01:02.920
So we have to evaluate
these exponents

00:01:02.920 --> 00:01:06.270
before we multiply anything or
before we subtract anything.

00:01:06.270 --> 00:01:09.270
So the one exponent we'd have
to evaluate is 3 squared.

00:01:09.270 --> 00:01:10.020
So let's remember.

00:01:10.020 --> 00:01:11.610
3 to the first is just 3.

00:01:11.610 --> 00:01:14.080
It's just 3 times itself once.

00:01:14.080 --> 00:01:14.776
So it's just 3.

00:01:14.776 --> 00:01:21.020
3 squared is equal to 3 times
3, 3 multiplied by itself twice.

00:01:21.020 --> 00:01:22.930
That's equal to 9.

00:01:22.930 --> 00:01:27.530
3 to the third power is
equal to 3 times 3 times 3.

00:01:27.530 --> 00:01:30.400
Or you could view it
as 3 squared times 3.

00:01:30.400 --> 00:01:31.870
So it'll be 9.

00:01:31.870 --> 00:01:32.840
3 times 3 is 9.

00:01:32.840 --> 00:01:35.580
9 times 3 is equal to 27.

00:01:35.580 --> 00:01:41.740
3 to the fourth is equal to
3 times 3 times 3 times 3.

00:01:41.740 --> 00:01:44.010
So 3 times 3 is 9.

00:01:44.010 --> 00:01:45.360
3 times 3 is 9.

00:01:45.360 --> 00:01:47.920
So it's going to be the
same thing as 9 times 9.

00:01:47.920 --> 00:01:49.875
So this is going
to be equal to 81.

00:01:49.875 --> 00:01:51.500
So we now know what
3 to the fourth is.

00:01:51.500 --> 00:01:52.650
We know what 3 squared is.

00:01:52.650 --> 00:01:54.150
Let's just put it
in the expression.

00:01:54.150 --> 00:01:58.700
So this is going to be equal
to 5 times 3 to the fourth.

00:01:58.700 --> 00:02:01.440
3 to the fourth is 81.

00:02:01.440 --> 00:02:06.390
So 5 times 81 minus 3 squared.

00:02:06.390 --> 00:02:08.130
And we have 3 squared
right over here.

00:02:08.130 --> 00:02:10.150
It is equal to 9.

00:02:10.150 --> 00:02:12.670
5 times 81 minus 9.

00:02:12.670 --> 00:02:14.540
Let's figure out
what 5 times 81 is.

00:02:14.540 --> 00:02:17.260
So 81 times 5.

00:02:17.260 --> 00:02:18.930
1 times 5 is 5.

00:02:18.930 --> 00:02:21.850
8 times 5 is 40.

00:02:21.850 --> 00:02:25.350
So this right over here is 405.

00:02:25.350 --> 00:02:29.270
So it becomes 405 minus 9.

00:02:29.270 --> 00:02:32.530
So that is going to be equal
to-- if we were subtracting 10,

00:02:32.530 --> 00:02:34.122
it would be 395.

00:02:34.122 --> 00:02:35.920
But we're subtracting
one less than that.

00:02:35.920 --> 00:02:39.160
So it's 396.

00:02:39.160 --> 00:02:41.160
And we're done.


Exponent properties 1

WEBVTT
Kind: captions
Language: en

00:00:00.700 --> 00:00:05.110
Simplify 3a times a to
the fifth times a squared.

00:00:05.110 --> 00:00:07.950
So the exponent
property we can use here

00:00:07.950 --> 00:00:11.210
is if we have the same
base, in this case, it's a.

00:00:11.210 --> 00:00:13.110
If we have it raised
to the x power,

00:00:13.110 --> 00:00:16.400
we're multiplying it
by a to the y power,

00:00:16.400 --> 00:00:20.870
then this is just going to
be equal to a to the x plus y

00:00:20.870 --> 00:00:21.370
power.

00:00:21.370 --> 00:00:23.411
And we'll think about why
that works in a second.

00:00:23.411 --> 00:00:24.730
So let's just apply it here.

00:00:24.730 --> 00:00:28.040
Let's start with the a to
the fifth times a squared.

00:00:28.040 --> 00:00:30.490
So if we just apply
this property over here,

00:00:30.490 --> 00:00:35.940
this will result in a to
the fifth plus two-th power.

00:00:35.940 --> 00:00:39.480
So that's what those guys
reduce to, or simplify to.

00:00:39.480 --> 00:00:43.700
And of course, we still
have the 3a out front.

00:00:43.700 --> 00:00:45.950
Now what I want to do is
take a little bit of an aside

00:00:45.950 --> 00:00:47.930
and realize why this worked.

00:00:47.930 --> 00:00:51.760
Let's think about what a to the
fifth times a squared means.

00:00:51.760 --> 00:00:55.880
A to the fifth
literally means a times

00:00:55.880 --> 00:01:00.170
a times a times a times a.

00:01:00.170 --> 00:01:05.890
Now, a squared literally
means a times a.

00:01:05.890 --> 00:01:08.120
And we're multiplying
the two times each other.

00:01:08.120 --> 00:01:11.796
So we're multiplying these
five a's times these two a's.

00:01:11.796 --> 00:01:12.920
And what have we just done?

00:01:12.920 --> 00:01:15.110
We're multiplying a
times itself five times,

00:01:15.110 --> 00:01:16.375
and then another two times.

00:01:19.590 --> 00:01:21.560
We are multiplying
a times itself.

00:01:21.560 --> 00:01:22.560
So let me make it clear.

00:01:22.560 --> 00:01:24.950
This over here is
a to the fifth.

00:01:24.950 --> 00:01:27.410
This over here is a squared.

00:01:27.410 --> 00:01:29.020
When you multiply
the two, you're

00:01:29.020 --> 00:01:34.870
multiplying a by itself
itself seven times.

00:01:34.870 --> 00:01:36.180
5 plus 2.

00:01:36.180 --> 00:01:40.680
So this is a to the seventh
power. a to the 5 plus 2 power.

00:01:40.680 --> 00:01:45.945
So this simplifies to 3a
times a to the seventh power.

00:01:48.750 --> 00:01:51.460
Now you might say, how do I
apply the property over here?

00:01:51.460 --> 00:01:52.790
What is the exponent on the a?

00:01:52.790 --> 00:01:55.490
And remember, if I just
have an a over here,

00:01:55.490 --> 00:01:58.920
this is equivalent to
a to the first power.

00:01:58.920 --> 00:02:02.830
So I can rewrite 3a is 3
times a to the first power.

00:02:02.830 --> 00:02:05.710
And now it maybe makes
it a little bit clearer.

00:02:05.710 --> 00:02:08.804
A to the first power-- and
the association property

00:02:08.804 --> 00:02:10.720
of multiplication, I can
do the multiplication

00:02:10.720 --> 00:02:13.100
of the a's before I
worry about the 3's.

00:02:13.100 --> 00:02:15.940
So I can multiply
these two guys first.

00:02:15.940 --> 00:02:17.992
So a to the first times
a to the seventh-- I

00:02:17.992 --> 00:02:20.450
just have to add the exponents
because I have the same base

00:02:20.450 --> 00:02:22.020
and I'm taking the
product-- that's

00:02:22.020 --> 00:02:24.250
going to be a to
the eighth power.

00:02:24.250 --> 00:02:27.110
And I still have
this 3 out front.

00:02:27.110 --> 00:02:29.840
So 3a times a to
the fifth times a

00:02:29.840 --> 00:02:35.530
squared simplifies to
3a to the eighth power.


How to solve equations of the form x/a = b

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:03.370
Solve for x and
check your solution.

00:00:03.370 --> 00:00:07.920
We have x divided
by 3 is equal to 14.

00:00:07.920 --> 00:00:11.320
So to solve for x, to figure
out what the variable x must

00:00:11.320 --> 00:00:14.135
be equal to, we really
just have to isolate it

00:00:14.135 --> 00:00:15.760
on the left-hand side
of this equation.

00:00:15.760 --> 00:00:16.884
It's already sitting there.

00:00:16.884 --> 00:00:19.490
We have x divided
by 3 is equal to 14.

00:00:19.490 --> 00:00:24.960
We could also write this
as 1/3 x is equal to 14.

00:00:24.960 --> 00:00:27.730
Obviously, x times 1/3
is going to be x/3.

00:00:27.730 --> 00:00:28.840
These are equivalent.

00:00:28.840 --> 00:00:31.800
So how can we just end up with
an x on the left-hand side

00:00:31.800 --> 00:00:33.107
of either of these equations?

00:00:33.107 --> 00:00:34.440
These are really the same thing.

00:00:34.440 --> 00:00:36.370
Or another way, how
can we just have a 1

00:00:36.370 --> 00:00:38.990
in front of the x, a 1x,
which is really just saying

00:00:38.990 --> 00:00:40.650
x over here?

00:00:40.650 --> 00:00:42.860
Well, I'm dividing
it by 3 right now.

00:00:42.860 --> 00:00:46.420
So if I were to multiply both
sides of this equation by 3,

00:00:46.420 --> 00:00:47.680
that would isolate the x.

00:00:47.680 --> 00:00:50.100
And the reason that would
work is if I multiply this

00:00:50.100 --> 00:00:54.140
by 3 over here, I'm multiplying
by 3 and dividing by 3.

00:00:54.140 --> 00:00:55.310
That's equivalent.

00:00:55.310 --> 00:00:57.720
That's equivalent to
multiplying or dividing by 1.

00:00:57.720 --> 00:00:58.995
These guys cancel out.

00:00:58.995 --> 00:01:00.870
Remember, if you do it
to the left-hand side,

00:01:00.870 --> 00:01:03.004
you also have to do it
to the right-hand side.

00:01:03.004 --> 00:01:04.879
And actually, I'll do
both of these equations

00:01:04.879 --> 00:01:06.545
at the same time,
because they're really

00:01:06.545 --> 00:01:08.470
the exact same equation.

00:01:08.470 --> 00:01:10.970
So what are we going to get
over here on the left-hand side?

00:01:10.970 --> 00:01:15.350
3 times anything divided by 3
is going to be that anything.

00:01:15.350 --> 00:01:17.580
We're just going to
have an x left over

00:01:17.580 --> 00:01:19.050
on the left-hand side.

00:01:19.050 --> 00:01:22.100
And on the right-hand
side, what's 14 times 3?

00:01:22.100 --> 00:01:27.770
3 times 10 is 30,
3 times 4 is 12.

00:01:27.770 --> 00:01:29.510
So it's going to be 42.

00:01:29.510 --> 00:01:31.550
So we get x is equal to 42.

00:01:31.550 --> 00:01:33.120
And the same thing
would happen here.

00:01:33.120 --> 00:01:36.250
3 times 1/3 is just 1.

00:01:36.250 --> 00:01:39.820
So you get 1x is equal to
14 times 3, which is 42.

00:01:39.820 --> 00:01:41.250
Now let's just check our answer.

00:01:41.250 --> 00:01:44.060
Let's substitute 42 into
our original equation.

00:01:44.060 --> 00:01:52.190
So we have 42 in place for
x over 3 is equal to 14.

00:01:52.190 --> 00:01:54.370
So what's 42 divided by 3?

00:01:54.370 --> 00:01:57.210
And we could do a little
bit of-- I guess we call it

00:01:57.210 --> 00:01:58.200
medium-long division.

00:01:58.200 --> 00:01:59.690
It's not really long division.

00:01:59.690 --> 00:02:01.210
3 into 4.

00:02:01.210 --> 00:02:02.580
3 goes into 4 one time.

00:02:02.580 --> 00:02:04.370
1 times 3 is 3.

00:02:04.370 --> 00:02:05.250
You subtract.

00:02:05.250 --> 00:02:07.100
4 minus 3 is 1.

00:02:07.100 --> 00:02:08.370
Bring down the 2.

00:02:08.370 --> 00:02:13.110
3 goes into 12 four times.

00:02:13.110 --> 00:02:14.710
3 goes into 42 14 times.

00:02:14.710 --> 00:02:19.230
So this right over
here simplifies to 14.

00:02:19.230 --> 00:02:22.640
And it all checks
out, so we're done.


Example of plotting a simple inequality on a number line

WEBVTT
Kind: captions
Language: en

00:00:00.780 --> 00:00:03.930
Graph x is less than 4.

00:00:03.930 --> 00:00:07.350
So let's draw ourselves
a number line over here.

00:00:07.350 --> 00:00:09.755
So let me draw a number line.

00:00:12.580 --> 00:00:20.220
I'll start here at 0,
so 0, 1, 2, 3, 4, 5.

00:00:20.220 --> 00:00:21.350
And we could go below 0.

00:00:21.350 --> 00:00:26.360
We'd have negative 1, negative
2, negative 3, negative 4.

00:00:26.360 --> 00:00:27.590
I could keep going.

00:00:27.590 --> 00:00:31.950
Now, we want to graph all of
the x's that are less than 4,

00:00:31.950 --> 00:00:33.370
but we're not including 4.

00:00:33.370 --> 00:00:35.250
It's not less than
or equal to 4.

00:00:35.250 --> 00:00:36.950
It's just less than 4.

00:00:36.950 --> 00:00:39.750
And to show that we're not
going to include 4, what we're

00:00:39.750 --> 00:00:42.890
going to do is we're going
to draw a circle around 4.

00:00:42.890 --> 00:00:45.400
So this shows us that
we're not including 4.

00:00:45.400 --> 00:00:49.125
If we were including 4, I
would make that a solid dot.

00:00:49.125 --> 00:00:52.960
And to show that we're going to
do all the values less than 4,

00:00:52.960 --> 00:00:55.060
we want to shade
in the number line

00:00:55.060 --> 00:00:59.045
below 4, going down
from 4, just like that.

00:01:02.380 --> 00:01:06.050
And then we can just shade
in the arrow just like that.

00:01:06.050 --> 00:01:09.657
So this right here is all
of the values less than 4.

00:01:09.657 --> 00:01:10.740
And you could test it out.

00:01:10.740 --> 00:01:13.080
Take any value
where there's blue.

00:01:13.080 --> 00:01:15.510
So there's blue over
here, negative 2.

00:01:15.510 --> 00:01:20.830
Negative 2 is
definitely less than 4.

00:01:20.830 --> 00:01:22.670
If you take this value
right here, this 2,

00:01:22.670 --> 00:01:24.526
it's definitely less than 4.

00:01:24.526 --> 00:01:27.720
4 is not included because
4 is not less than 4.

00:01:27.720 --> 00:01:28.720
It's equal to 4.

00:01:28.720 --> 00:01:33.200
5 is not included because
5 is not less than 4.


Example of solving a one-step equation

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:03.180
Solve for a and
check your solution.

00:00:03.180 --> 00:00:06.054
And we have a plus
5 is equal to 54.

00:00:06.054 --> 00:00:07.470
Now, all this is
saying is that we

00:00:07.470 --> 00:00:10.310
have some numbers,
some variable a.

00:00:10.310 --> 00:00:13.484
And if I add 5 to
it, I will get 54.

00:00:13.484 --> 00:00:15.400
And you might be able
to do this in your head.

00:00:15.400 --> 00:00:17.820
But we're going to do it a
little bit more systematically.

00:00:17.820 --> 00:00:19.570
Because that'll be
helpful for you when we

00:00:19.570 --> 00:00:21.070
do more complicated problems.

00:00:21.070 --> 00:00:23.361
So in general, whenever you
have an equation like this,

00:00:23.361 --> 00:00:24.930
we want to have the variable.

00:00:24.930 --> 00:00:29.670
We want this a all by itself
on one side of the equation.

00:00:29.670 --> 00:00:31.360
We want to isolate it.

00:00:31.360 --> 00:00:32.819
It's already on
the left-hand side,

00:00:32.819 --> 00:00:34.360
so let's try to get
rid of everything

00:00:34.360 --> 00:00:35.490
else on the left-hand side.

00:00:35.490 --> 00:00:37.490
Well, the only other thing
on the left-hand side

00:00:37.490 --> 00:00:38.580
is this positive 5.

00:00:38.580 --> 00:00:42.440
Well, the best way to get rid
of a plus 5, or a positive 5,

00:00:42.440 --> 00:00:44.910
is to subtract 5.

00:00:44.910 --> 00:00:46.300
So let's subtract 5.

00:00:46.300 --> 00:00:50.110
But remember, this says
a plus 5 is equal to 54.

00:00:50.110 --> 00:00:52.180
If we want the
equality to still hold,

00:00:52.180 --> 00:00:54.430
anything we do to the left-hand
side of this equation,

00:00:54.430 --> 00:00:57.550
we have to do to the right
side of the equation.

00:00:57.550 --> 00:01:00.660
So we also have to
subtract 54 from the right.

00:01:00.660 --> 00:01:03.090
So we have a plus 5 minus 5.

00:01:03.090 --> 00:01:05.030
Well, that's just
going to be a plus 0.

00:01:05.030 --> 00:01:08.080
Because if you add 5 and you
subtract 5, they cancel out.

00:01:08.080 --> 00:01:10.400
So a plus 0 is just a.

00:01:10.400 --> 00:01:14.520
And then 54 minus 5, that is 49.

00:01:14.520 --> 00:01:15.220
And we're done.

00:01:15.220 --> 00:01:16.975
We have solved for a.

00:01:16.975 --> 00:01:19.460
A is equal to 49.

00:01:19.460 --> 00:01:20.810
And now we can check it.

00:01:20.810 --> 00:01:24.270
And we can check it by
just substituting 49 back

00:01:24.270 --> 00:01:27.100
for a in our original equation.

00:01:27.100 --> 00:01:29.600
So instead of writing a
plus 5 is equal to 54,

00:01:29.600 --> 00:01:32.040
let's see if 49 plus
5 is equal to 54.

00:01:32.040 --> 00:01:34.010
So we're just
substituting it back in.

00:01:34.010 --> 00:01:39.920
49, 49 plus-- let me do that
in that same shade of green.

00:01:39.920 --> 00:01:43.730
49 plus 5 is equal to 54.

00:01:43.730 --> 00:01:45.270
We're trying to check this.

00:01:45.270 --> 00:01:48.210
49 plus 5 is 54.

00:01:48.210 --> 00:01:51.140
And that, indeed,
is equal to 54.

00:01:51.140 --> 00:01:54.080
So it all checks out.


Example 1: Variables on both sides

WEBVTT
Kind: captions
Language: en

00:00:00.680 --> 00:00:03.300
Solve for y and
check your solution.

00:00:03.300 --> 00:00:07.772
20 minus 7y is
equal to 6y minus 6.

00:00:07.772 --> 00:00:09.230
So what I really
want to just do is

00:00:09.230 --> 00:00:12.154
isolate all of the y's on
one side of this equation.

00:00:12.154 --> 00:00:14.320
And then if there's some
coefficient in front of it,

00:00:14.320 --> 00:00:17.070
I can divide both sides
by that coefficient.

00:00:17.070 --> 00:00:19.730
And I could isolate the
y on either the left side

00:00:19.730 --> 00:00:21.420
or the right side
of the equation.

00:00:21.420 --> 00:00:23.660
Actually, let me
do it both ways.

00:00:23.660 --> 00:00:27.434
So first, it looks like it'll
be a little bit simpler if I

00:00:27.434 --> 00:00:28.850
isolate it on the
right hand side,

00:00:28.850 --> 00:00:30.308
and I'll show you
that in a second.

00:00:30.308 --> 00:00:33.140
So if I want to put all of the
Y's on the right hand side,

00:00:33.140 --> 00:00:35.510
how could I get rid of the
Y's on the left hand side?

00:00:35.510 --> 00:00:39.110
I have a minus 7y, or you
could use the negative 7y.

00:00:39.110 --> 00:00:42.480
I can add 7y to both
sides of this equation.

00:00:42.480 --> 00:00:46.172
So I'm just adding
7y to both sides.

00:00:46.172 --> 00:00:47.880
Then the left hand
side of the equation--

00:00:47.880 --> 00:00:50.550
we have 20 minus 7y plus 7y.

00:00:50.550 --> 00:00:51.800
Those are going to cancel out.

00:00:51.800 --> 00:00:54.039
You're going to be
just left with 20.

00:00:54.039 --> 00:00:55.830
And on the right hand
side of the equation,

00:00:55.830 --> 00:01:01.160
you're going to have 6y
plus 7y, which is 13y.

00:01:01.160 --> 00:01:03.720
And then you have your minus 6.

00:01:03.720 --> 00:01:07.022
And I want only
y's, or multiples

00:01:07.022 --> 00:01:08.980
of y's, on the right hand
side of the equation.

00:01:08.980 --> 00:01:12.320
So I need to get rid of
this minus 6 over here.

00:01:12.320 --> 00:01:15.550
So let's add 6 to both
sides of this equation.

00:01:18.540 --> 00:01:22.830
And then on the right hand
side, I have 13y minus 6 plus 6.

00:01:22.830 --> 00:01:26.170
That's just 13y-- got
rid of the minus 6.

00:01:26.170 --> 00:01:27.600
And I have 20 plus 6.

00:01:27.600 --> 00:01:29.570
So that is 26.

00:01:29.570 --> 00:01:32.280
So I get 26 is equal to 13y.

00:01:32.280 --> 00:01:34.130
I've isolated the y on
the right hand side,

00:01:34.130 --> 00:01:36.370
although it's being
multiplied by 13.

00:01:36.370 --> 00:01:38.890
So if I just want a y
on the right hand side

00:01:38.890 --> 00:01:43.690
or 1y on the right hand side,
I can divide both sides by 13.

00:01:43.690 --> 00:01:45.420
Or the equivalent
statement would

00:01:45.420 --> 00:01:48.380
be to multiply both sides
by 1 over 13-- same thing.

00:01:48.380 --> 00:01:52.800
So let's divide both sides
of this equation by 13.

00:01:52.800 --> 00:01:56.020
On the right hand side,
13 times y divided by 13.

00:01:56.020 --> 00:01:57.880
Well, that's just going to be y.

00:01:57.880 --> 00:01:59.240
So this is just y.

00:01:59.240 --> 00:02:03.160
And then 26 divided
by 13-- that is 2.

00:02:03.160 --> 00:02:05.309
So our solution we got
is y is equal to 2.

00:02:05.309 --> 00:02:06.600
Now I want to take a step back.

00:02:06.600 --> 00:02:10.370
Why did I say it would
be a little bit easier

00:02:10.370 --> 00:02:13.230
to put the y's on the right
hand side of the equation?

00:02:13.230 --> 00:02:16.060
Well, because that's the way
I got a positive coefficient

00:02:16.060 --> 00:02:17.010
in terms of y.

00:02:17.010 --> 00:02:18.170
I got 13y.

00:02:18.170 --> 00:02:20.912
If I had subtracted
6y from both sides,

00:02:20.912 --> 00:02:23.370
I would have got a negative 13
on the left hand side, which

00:02:23.370 --> 00:02:24.680
would have been a
legitimate thing to do.

00:02:24.680 --> 00:02:26.830
Let me just show you
what I'm talking about.

00:02:26.830 --> 00:02:34.180
So I could have done the
problem this way-- 20 minus

00:02:34.180 --> 00:02:38.520
7y is equal to 6y minus 6.

00:02:38.520 --> 00:02:40.920
Now let me just subtract
6y from both sides.

00:02:40.920 --> 00:02:45.590
So minus 6y, or negative
6y, from both sides.

00:02:45.590 --> 00:02:49.850
The left hand side over
here is 20 minus 13y.

00:02:49.850 --> 00:02:52.370
The right hand side--
I got rid of that 6y.

00:02:52.370 --> 00:02:56.550
I just have the minus 6 here,
or it's just a negative 6 now.

00:02:56.550 --> 00:02:59.980
And now I can subtract
20 from both sides

00:02:59.980 --> 00:03:03.300
to isolate the y's on
the left hand side.

00:03:03.300 --> 00:03:06.950
So I'm subtracting
20 from both sides.

00:03:06.950 --> 00:03:09.220
I'm doing that to
get rid of this 20.

00:03:09.220 --> 00:03:11.860
And so on the left hand
side, I have a negative 13y,

00:03:11.860 --> 00:03:16.290
and on the right hand
side, I have a negative 26.

00:03:16.290 --> 00:03:18.320
Now I want to divide both
sides of this equation

00:03:18.320 --> 00:03:25.890
by negative 13,
and what do I get?

00:03:25.890 --> 00:03:28.320
Negative 13 times something
divided by negative 13

00:03:28.320 --> 00:03:29.490
is just that something.

00:03:29.490 --> 00:03:30.580
I just get a y.

00:03:30.580 --> 00:03:34.320
And then negative 26
divided by negative 13 is 2.

00:03:34.320 --> 00:03:35.880
So I did it two different ways.

00:03:35.880 --> 00:03:38.270
Over here, I isolated the
y's on the left hand side.

00:03:38.270 --> 00:03:40.074
Over here, I did it on
the right hand side.

00:03:40.074 --> 00:03:41.490
I picked the right
hand side first

00:03:41.490 --> 00:03:42.970
so that everything
stays positive,

00:03:42.970 --> 00:03:45.669
but this is an equally
valid way to do it.

00:03:45.669 --> 00:03:47.710
Now the last thing I want
to do is just make sure

00:03:47.710 --> 00:03:49.579
that we got the right answer.

00:03:49.579 --> 00:03:51.370
We should probably feel
pretty comfortable.

00:03:51.370 --> 00:03:53.290
We got the same answer
doing it two different ways,

00:03:53.290 --> 00:03:54.456
but let's actually check it.

00:03:54.456 --> 00:03:56.630
Let's put y equal 2
back into the equation

00:03:56.630 --> 00:03:59.500
and verify that we
get the equality.

00:03:59.500 --> 00:04:03.130
So we have 20 minus 7y is 2.

00:04:03.130 --> 00:04:11.930
So 7 times 2 should be
equal to 6 times 2 minus 6.

00:04:11.930 --> 00:04:14.970
So this should be 20 minus 14.

00:04:14.970 --> 00:04:17.810
I want to do multiplication
first-- order of operations.

00:04:17.810 --> 00:04:21.850
So this is 20 minus 14
should be equal to 6 times

00:04:21.850 --> 00:04:24.166
2 is 12 minus 6.

00:04:24.166 --> 00:04:26.224
20 minus 14 is 6.

00:04:26.224 --> 00:04:28.440
12 minus 6 is 6.

00:04:28.440 --> 00:04:33.010
6 is, indeed, equal to
6, so it all checks out.