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All right, Shmoopers, it's about time we had...the talk. Sometimes, when two polynomials love...

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each other very much, well, they decide to multiply. Now...

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we're gonna explain to you how that happens. We'll try to leave out most of the gory details. All right,

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so say you have two polyn—and get your minds out of the gutter, people. All right, say you have two polynomials...

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with multiple terms. Like there's an x cubed in there, and a couple of x squareds,

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some x's, and so on. Something like, well, these two. Well, to multiply...

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them, we have to be sure to multiply every term in the first polynomial by every term in the second polynomial.

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Okay? That's this, times this times this times this. Yeah, like that.

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It's a delicate process. We miss a single term, and we could end up with a polynomial...

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with three heads. It's been known to happen—ever see the people who live down the river? Yeah.

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Okay, so we start with the 2x cubed in the first...

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polynomial, and multiply it by every term in the second. So, 2x cubed times 4x squared is...

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8x to the fifth. Add these things. 2x cubed times...

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3x is 6x to the fourth.

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Got it? So x is just x to the first, and we add it with x cubed...

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And then 2x cubed times 2 is 4x cubed.

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All right, then we move on to the second term, and the first polynomial, and do the same spiel.

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3x squared times 4x squared is 12x to the fourth.

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3x squared times 3x is 9x to the third.

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And 3x squared times 2 is 6x squared.

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Remember when you see an x, remember that's to to the one,

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and when you're multiplying exponents, you add 'em.

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All right, now for the third term in the first polynomial—this thing. Now don't forget the minus...

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sign. That's gonna turn all our answers to your negative.

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Negative x times 4x squared is negative 4x cubed,

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negative x times 3x is negative 3x squared, and...

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negative x times 2 is negative 2x.

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Okay! Last one.

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7 times 4x squared is 28x squared.

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7 times 3x is 21x,

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and 7 times 2 is 14. Okay, great!

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So now, we have....this. Yeuch, what a mess.

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No wonder people cringe when they, uh, have the talk and hear how this stuff works.

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But we can make it much, much prettier. We just have to combine like terms.

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We only have one term to the fifth power, so we can leave that one as it is.

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Then we add these two guys, then these three, then...

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these three, then these two, and this last one gets...

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left as is, too. But we finally just string them all together,

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to get 8x to the fifth, plus 18x to the fourth, plus...

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9x to the third, plus 31x squared, plus 19x, plus 14...

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And a partridge in a pear tree. Yeah, like that. All right, so...

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a little more complicated than their parents, but you can still see some similarities.

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When multiplying three or more polynomials? Same deal. Just takes...

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a little more time. You'd multiply the first two polynomials the way you usually would,

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and then take that answer and multiply it by the next polynomial in line.

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Oh, be careful, though, it's only legal in some parts of the red states.

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