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# 特殊的二項式乘法結果 (英): (a+b)(a-b)

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- And now I want to do a bunch of examples dealing with
- probably the two most typical types of polynomial
- multiplication that you'll see, definitely, in algebra.
- And the first is just squaring a binomial.
- So if I have x plus 9 squared, I know that your temptation is
- going to say, oh, isn't that x squared plus 9 squared?
- And I'll say, no, it isn't.
- You have to resist every temptation on the
- planet to do this.
- It is not x squared plus 9 squared.
- Remember, x plus 9 squared, this is equal to x plus 9,
- times x plus 9.
- This is a multiplication of this binomial times itself.
- You always need to remember that.
- It's very tempting to think that it's just x squared plus
- 9 squared, but no, you have to expand it out.
- And now that we've expanded it out, we can use some of the
- skills we learned in the last video to actually multiply it.
- And just to show you that we can do it in the way that we
- multiplied the trinomial last time, let's multiply x plus 9,
- times x plus a magenta 9.
- And I'm doing it this way just to show you when I'm
- multiplying by this 9 versus this x.
- But let's just do it.
- So we go 9 times 9 is 81.
- Put it in the constants' place.
- 9 times x is 9x.
- Then we have-- go switch to this x term-- we have a yellow
- x. x times 9x is 9x.
- Put it in the first degree space.
- x times x is x squared.
- And then we add everything up.
- And we get x squared plus 18x plus 81.
- So this is equal to x squared plus 18x plus 81.
- Now you might see a little bit of a pattern here, and I'll
- actually make the pattern explicit in a second.
- But when you square a binomial, what happened?
- You have x squared.
- You have this x times this x, gives you x squared.
- You have the 9 times the 9, which is 81.
- And then you have this term here which is 18x.
- How did we get that 18x?
- Well, we multiplied this x times 9 to get 9x, and then we
- multiplied this 9 times x to get another 9x.
- And then we added the two right here to get 18x.
- So in general, whenever you have a squared binomial-- let
- me do it this way.
- I'll do it in very general terms. Let's say we have a
- plus b squared.
- Let me multiply it this way again, just to give you the
- hang of it.
- This is equal to a plus b, times a plus-- I'll do a green
- b right there.
- So we have to b times b is b squared.
- Let's just assume that this is a constant term.
- I'll put it in the b squared right there.
- I'm assuming this is constant.
- So this would be a constant, this would be
- analogous to our 81.
- a is a variable that we-- actually let me change that up
- even better.
- Let me make this into x plus b squared, and we're assuming b
- is a constant.
- So it would be x plus b, times x plus a green b, right there.
- So assuming b's a constant, b times b is b squared.
- b times x is bx.
- And then we'll do the magenta x.
- x times b is bx.
- And then x times x is x squared.
- So when you add everything, you're left with x squared
- plus 2bx, plus b squared.
- So what you see is, the end product, what you have when
- you have x plus b squared, is x squared, plus 2 times the
- product of x and b, plus b squared.
- So given that pattern, let's do a bunch more of these.
- And I'm going to do it the fast way.
- So 3x minus 7 squared.
- Let's just remember what I told you.
- Just don't remember it, in the back of your mind, you should
- know why it makes sense.
- If I were to multiply this out, do the distributive
- property twice, you know you'll get the same answer.
- So this is going to be equal to 3x squared, plus 2 times
- 3x, times negative 7.
- Right?
- We know that it's 2 times each the product of these terms,
- plus negative 7 squared.
- And if we use our product rules here, 3x squared is the
- same thing as 9x squared.
- This right here, you're going to have a 2 times a 3, which
- is 6, times a negative 7, which is negative 42x.
- And then a negative 7 squared is plus 49.
- That was the fast way.
- And just to make sure that I'm not doing something bizarre,
- let me do it the slow way for you.
- 3x minus 7, times 3x minus 7.
- Negative 7 times negative 7 is positive 49.
- Negative 7 times 3x is negative 21x.
- 3x times negative 7 is negative 21x.
- 3x times 3x is 9 x squared.
- Scroll to the left a little bit.
- Add everything.
- You're left with 9x squared, minus 42x, plus 49.
- So we did indeed get the same answer.
- Let's do one more, and we'll do it the fast way.
- So if we have 8x minus 3-- actually, let me do one which
- has more variables in it.
- Let's say we had 4x squared plus y squared, and we wanted
- to square that.
- Well, same idea.
- This is going to be equal to this term squared, 4x squared,
- squared, plus 2 times the product of both terms, 2 times
- 4x squared times y squared, plus y
- squared, this term, squared.
- And what's this going to be equal to?
- This is going to be equal to 16-- right, 4 squared is 16--
- x squared, squared, that's 2 times 2, so it's x to the
- fourth power.
- And then plus, 2 times 4 times 1, that's
- 8x squared y squared.
- And then y squared, squared, is y to the fourth.
- Now, we've been dealing with squaring a binomial.
- The next example I want to show you is when I take the
- product of a sum and a difference.
- And this one actually comes out pretty neat.
- So I'm going to do a very general one for you.
- Let's just do a plus b, times a minus b.
- So what's this going to be equal to?
- This is going to be equal to a times a-- let me make these
- actually in different colors-- so a minus b, just like that.
- So it's going to be this green a times this magenta a, a
- times a, plus, or maybe I should say minus, the green a
- times this b.
- I got the minus from right there.
- And then we're going to have the green b, so plus the green
- b times the magenta a.
- I'm just multiplying every term by every term.
- And then finally minus the green b-- that's where the
- minus is coming from-- minus the green b
- times the magenta b.
- And what is this going to be equal to?
- This is going to be equal to a squared, and then
- this is minus ab.
- This could be rewritten as plus ab, and then we have
- minus b squared.
- These right here cancel out, minus ab plus ab, so you're
- just left with a squared minus b squared.
- Which is a really neat result because it
- really simplifies things.
- So let's use that notion to do some multiplication.
- So if we say 2x minus 1, times 2x plus 1.
- Well, these are the same thing.
- The 2x plus 1, you could view this as, if you like, a plus
- b, and the 2x minus 1, you can view it as a minus b, where
- this is a, and that b is 1.
- This is b.
- That is a.
- Just using this pattern that we figured out just now.
- So what is this going to be equal to?
- It's going to be a squared, it's going to be 2x squared,
- minus b squared, minus 1 squared.
- 2x squared is 4x squared.
- 1 squared is just 1, so minus 1.
- So it's going to be 4x squared minus 1.
- Let's do one more of these, just to really
- hit the point home.
- I'll just focus on multiplication right now.
- If I have 5a minus 2b, and I'm multiplying that
- times 5a plus 2b.
- And remember, this only applies when I have at a
- product of a sum and a difference.
- That's the only time that I can use this.
- And I've shown you why.
- And if you're ever in doubt, just multiply it out.
- It'll take you a little bit longer.
- And you'll see the terms canceling out.
- You can't do this for just any binomial multiplication.
- You saw that earlier in the video, when we were
- multiplying, when we were taking squares.
- So this is going to be, using the pattern, it's going to be
- 5a squared minus 2b squared, which is equal to 25 a squared
- minus 4b squared.
- And, well, I'll leave it there, and I'll see you in the
- next video.

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