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Multiplying binomials by polynomials (old)

An old video where Sal gives several examples of polynomial multiplication. Created by Sal Khan.

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Video transcript

In this video I'm just going to multiply a ton of polynomials, and hopefully that'll give you enough exposure to feel confident when you have to multiply any for yourselves. Let's start with a fairly simple problem. Let's say we just want to multiply 2x times 4x minus 5. Well, we just straight up use the distributive property here. And really, when we do all of these polynomial multiplications, all we're doing is the distributive property repeatedly. But let's just do the distributive property here. This is 2x times 4x, plus 2x times negative 5. Or we could say negative 5 times 2x. So you'd say, minus 5 times 2x. All I did is distribute the 2x. This first term is going to be equal to-- we can multiply the coefficients. Remember, 2x times 4x is the same thing as-- you can rearrange the order of multiplication. This is the same thing as 2 times 4, times x times x. Which is the same thing as 8 times x squared. Remember, x to the 1, times x to the 1, add the exponents. I mean, you know x times x is x squared. So this first term is going to be 8x squared. And the second term, negative 5 times 2 is negative 10x. Not too bad. Let's do a slightly more involved one. Let's say we had 9x to the third power, times 3x squared, minus 2x, plus 7. So once again, we're just going to do the distributive property here. So we're going to multiply the 9x to the third times each of these terms. So 9x to the third times 3x squared. I'll write it out this time. In the next few, we'll start doing it a little bit in our heads. So this is going to be 9x to the third times 3x squared. And then we're going to have plus-- let me write it this way-- minus 2x times 9x to the third, and then plus 7 times 9x to the third. So sometimes I wrote the 9x to the third first, sometimes we wrote it later because I wanted this negative sign here. But it doesn't make a difference on the order that you're multiplying. So this first term here is going to be what? 9 times 3 is 27 times x to the-- we can add the exponents, we learned that in our exponent properties. This is x to the fifth power, minus 2 times 9 is 18x to the-- we have x to the 1, x to the third-- x to the fourth power. Plus 7 times 9 is 63x to the third. So we end up with this nice little fifth degree polynomial. Now let's do one where we are multiplying two binomials. And I'll show you what I mean in a second. This you're going to see very, very, very frequently in algebra. So let's say you have x minus 3, times x plus 2. And I actually want to show you that all we're doing here is the distributive property. So let me write it like this: times x plus 2. So let's just pretend that this is one big number here. And it is. You know, if you had x's, this would be some number here. So let's just distribute this onto each of these variables. So this is going to be x minus 3, times that green x, plus x minus 3, times that green 2. All we did is distribute the x minus 3. This is just the distributive property. Remember, if I had a times x plus 2, what would this be equal to? This would be equal to a times x plus a times 2. So over here, you can see when x minus 3 is the same thing as a, we're just distributing it. And now we would do the distributive property again. In this case, we're distributing the x now onto the x minus 3. We're going to distribute the 2 onto the x minus 3. You might be used to seeing the x on the other side, but either way, we're just multiplying it. So this is going to be-- I'll stay color coded. This is going to be x times x, minus 3 times x, plus x times 2-- I'm going through great pains to keep it color coded for you. I think it's helping-- minus 3 times 2. All I did is distribute the x and distribute the 2. And soon you're going to get used to this. We can do it in one step. You're actually multiplying every term in this one by every term in that one, and we'll figure out faster ways to do it in the future. But I really want to show you the idea here. So what's this going to equal? This is going to equal x squared. This right here is going to be minus 3x. This is going to be plus 2x. And then this right here is going to be minus 6. And so this is going to be x squared minus 3 of something, plus 2 of something, that's minus 1 of that something. Minus x, minus 6. We've multiplied those two. Now before we move on and do another problem, I want to show you that you can kind of do this in your head as well. You don't have to go through all of these steps. I just want to show you really that this is just the distributive property. The fast way of doing it, if you had x minus 3, times x plus 2, you literally just want to multiply every term here times each of these terms. So you'd say, this x times that x, so you'd have x squared. Then you'd have this x times that 2, so plus 2x. Then you'd have this minus 3 times that x, minus 3x. And then you have the minus 3, or the negative 3, times 2, which is negative 6. And so when you simplify, once again you get x squared minus x minus 6. And it takes a little bit of practice to really get used to it. Now the next thing I want to do-- and the principal is really the exact same way-- but I'm going to multiply a binomial times a trinomial, which many people find daunting. But we're going to see, if you just stay calm, it's not too bad. 3x plus 2, times 9x squared, minus 6x plus 4. Now you could do it the exact same way that we did the previous video. We could literally take this 3x plus 2, distribute it onto each of these three terms, multiply 3x plus 2 times each of these terms, and then you're going to distribute each of those terms into 3x plus 2. It would take a long time and in reality, you'll never do it quite that way. But you will get the same answer we're going to get. When you have larger polynomials, the easiest way I can think of to multiply, is kind of how you multiply long numbers. So we'll write it like this. 9x squared, minus 6, plus 4. And we're going to multiply that times 3x plus 2. And what I imagine is, when you multiply regular numbers, you have your ones' place, your tens' place, your hundreds' place. Here, you're going to have your constants' place, your first degree place, your second degree place, your third degree place, if there is one. And actually there will be in this video. So you just have to put things in their proper place. So let's do that. So you start here, you multiply almost exactly like you would do traditional multiplication. 2 times 4 is 8. It goes into the ones', or the constants' place. 2 times negative 6x is negative 12x. And we'll put a plus there. That was a plus 8. 2 times 9x squared is 18x squared, so we'll put that in the x squared place. Now let's do the 3x part. I'll do that in magenta, so you see how it's different. 3x times 4 is 12x, positive 12x. 3x times negative 6x, what is that? The x times the x is x squared, so it's going to go over here. And 3 times negative 6 is negative 18. And then finally 3x times 9x squared, the x times the x squared is x to the third power. 3 times 9 is 27. I wrote it in the x third place. And once again, you just want to add the like terms. So you get 8. There's no other constant terms, so it's just 8. Negative 12x plus 12x, these cancel out. 18x squared minus 18x squared cancel out, so we're just left over here with 27x to the third. So this is equal to 27x to the third plus 8. And we are done. And you can use this technique to multiply a trinomial times a binomial, a trinomial times a trinomial, or really, you know, you could have five terms up here. A fifth degree times a fifth degree. This will always work as long as you keep things in their proper degree place.