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In this video I'm just going to multiply a ton of

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polynomials, and hopefully that'll give you enough

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exposure to feel confident when you have to multiply any

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for yourselves.

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Let's start with a fairly simple problem.

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Let's say we just want to multiply 2x times 4x minus 5.

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Well, we just straight up use the

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distributive property here.

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And really, when we do all of these polynomial

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multiplications, all we're doing is the distributive

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property repeatedly.

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But let's just do the distributive property here.

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This is 2x times 4x, plus 2x times negative 5.

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Or we could say negative 5 times 2x.

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So you'd say, minus 5 times 2x.

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All I did is distribute the 2x.

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This first term is going to be equal to-- we can multiply the

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coefficients.

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Remember, 2x times 4x is the same thing as-- you can

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rearrange the order of multiplication.

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This is the same thing as 2 times 4, times x times x.

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Which is the same thing as 8 times x squared.

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Remember, x to the 1, times x to the 1, add the exponents.

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I mean, you know x times x is x squared.

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So this first term is going to be 8x squared.

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And the second term, negative 5 times 2 is negative 10x.

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Not too bad.

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Let's do a slightly more involved one.

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Let's say we had 9x to the third power, times 3x squared,

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minus 2x, plus 7.

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So once again, we're just going to do the distributive

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property here.

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So we're going to multiply the 9x to the third times each of

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these terms.

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So 9x to the third times 3x squared.

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I'll write it out this time.

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In the next few, we'll start doing it a

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little bit in our heads.

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So this is going to be 9x to the third times 3x squared.

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And then we're going to have plus-- let me write it this

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way-- minus 2x times 9x to the third, and then plus 7 times

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9x to the third.

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So sometimes I wrote the 9x to the third first, sometimes we

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wrote it later because I wanted this

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negative sign here.

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But it doesn't make a difference on the order that

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you're multiplying.

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So this first term here is going to be what?

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9 times 3 is 27 times x to the-- we can add the

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exponents, we learned that in our exponent properties.

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This is x to the fifth power, minus 2 times 9 is 18x to

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the-- we have x to the 1, x to the third-- x

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to the fourth power.

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Plus 7 times 9 is 63x to the third.

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So we end up with this nice little fifth degree

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polynomial.

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Now let's do one where we are multiplying two binomials.

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And I'll show you what I mean in a second.

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This you're going to see very, very, very

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frequently in algebra.

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So let's say you have x minus 3, times x plus 2.

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And I actually want to show you that all we're doing here

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is the distributive property.

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So let me write it like this: times x plus 2.

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So let's just pretend that this is one big number here.

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And it is.

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You know, if you had x's, this would be some number here.

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So let's just distribute this onto each of these variables.

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So this is going to be x minus 3, times that green x, plus x

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minus 3, times that green 2.

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All we did is distribute the x minus 3.

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This is just the distributive property.

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Remember, if I had a times x plus 2, what would

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this be equal to?

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This would be equal to a times x plus a times 2.

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So over here, you can see when x minus 3 is the same thing as

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a, we're just distributing it.

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And now we would do the distributive property again.

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In this case, we're distributing the x now onto

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the x minus 3.

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We're going to distribute the 2 onto the x minus 3.

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You might be used to seeing the x on the other side, but

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either way, we're just multiplying it.

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So this is going to be-- I'll stay color coded.

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This is going to be x times x, minus 3 times x, plus x times

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2-- I'm going through great pains to keep it

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color coded for you.

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I think it's helping-- minus 3 times 2.

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All I did is distribute the x and distribute the 2.

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And soon you're going to get used to this.

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We can do it in one step.

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You're actually multiplying every term in this one by

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every term in that one, and we'll figure out faster ways

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to do it in the future.

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But I really want to show you the idea here.

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So what's this going to equal?

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This is going to equal x squared.

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This right here is going to be minus 3x.

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This is going to be plus 2x.

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And then this right here is going to be minus 6.

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And so this is going to be x squared minus 3 of something,

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plus 2 of something, that's minus 1 of that something.

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Minus x, minus 6.

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We've multiplied those two.

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Now before we move on and do another problem, I want to

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show you that you can kind of do this in your head as well.

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You don't have to go through all of these steps.

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I just want to show you really that this is just the

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distributive property.

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The fast way of doing it, if you had x minus 3, times x

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plus 2, you literally just want to multiply every term

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here times each of these terms.

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So you'd say, this x times that x, so

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you'd have x squared.

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Then you'd have this x times that 2, so plus 2x.

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Then you'd have this minus 3 times that x, minus 3x.

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And then you have the minus 3, or the negative 3, times 2,

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which is negative 6.

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And so when you simplify, once again you get x squared

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minus x minus 6.

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And it takes a little bit of practice to

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really get used to it.

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Now the next thing I want to do-- and the principal is

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really the exact same way-- but I'm going to multiply a

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binomial times a trinomial, which

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many people find daunting.

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But we're going to see, if you just stay calm,

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it's not too bad.

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3x plus 2, times 9x squared, minus 6x plus 4.

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Now you could do it the exact same way that we did the

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previous video.

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We could literally take this 3x plus 2, distribute it onto

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each of these three terms, multiply 3x plus 2 times each

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of these terms, and then you're going to distribute

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each of those terms into 3x plus 2.

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It would take a long time and in reality, you'll never do it

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quite that way.

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But you will get the same answer we're going to get.

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When you have larger polynomials, the easiest way I

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can think of to multiply, is kind of how you

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multiply long numbers.

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So we'll write it like this.

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9x squared, minus 6, plus 4.

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And we're going to multiply that times 3x plus 2.

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And what I imagine is, when you multiply regular numbers,

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you have your ones' place, your tens' place, your

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hundreds' place.

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Here, you're going to have your constants' place, your

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first degree place, your second degree place, your

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third degree place, if there is one.

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And actually there will be in this video.

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So you just have to put things in their proper place.

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So let's do that.

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So you start here, you multiply almost exactly like

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you would do traditional multiplication.

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2 times 4 is 8.

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It goes into the ones', or the constants' place.

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2 times negative 6x is negative 12x.

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And we'll put a plus there.

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That was a plus 8.

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2 times 9x squared is 18x squared, so we'll put that in

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the x squared place.

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Now let's do the 3x part.

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I'll do that in magenta, so you see how it's different.

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3x times 4 is 12x, positive 12x.

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3x times negative 6x, what is that?

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The x times the x is x squared, so it's

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going to go over here.

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And 3 times negative 6 is negative 18.

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And then finally 3x times 9x squared, the x times the x

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squared is x to the third power.

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3 times 9 is 27.

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I wrote it in the x third place.

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And once again, you just want to add the like

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terms. So you get 8.

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There's no other constant terms, so it's just 8.

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Negative 12x plus 12x, these cancel out.

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18x squared minus 18x squared cancel out, so we're just left

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over here with 27x to the third.

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So this is equal to 27x to the third plus 8.

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And we are done.

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And you can use this technique to multiply a trinomial times

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a binomial, a trinomial times a trinomial, or really, you

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know, you could have five terms up here.

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A fifth degree times a fifth degree.

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This will always work as long as you keep things in their

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proper degree place.