Multiplying binomials intro

- [Voiceover] Let's see if we can figure out the product of x minus four and x plus seven. And we want to write that product in standard quadratic form which is just a fancy way of saying a form where you have some coefficient on the second degree term, a x squared plus some coefficient b on the first degree term plus the constant term. So this right over here would be standard quadratic form. So that's the form that we want to express this product in and encourage you to pause the video and try to work through it on your own. Alright, now let's work through this. And the key when we're multiplying two binomials like this, or actually when you're multiplying any polynomials, is just to remember the distributive property that we all by this point know quite well. So what we could view this is as is we could distribute this x minus four, this entire expression over the x and the seven. So we could say that this is the same thing as x minus four times x plus x minus four times seven. So let's write that. So x minus four times x, or we could write this as x times x minus four. That's distributing, or multiplying the x minus four times x that's right there. Plus seven times x minus four. Times x minus four. Notice all we did is distribute the x minus four. We took this whole thing and we multiplied it by each term over here. We multiplied x by x minus four and we multiplied seven by x minus four. Now, we see that we have these, I guess you can call them two seperate terms. And to simplify each of them, or to multiply them out, we just have to distribute. In this first we're going to have to distribute this blue x. And over here we have to distribute this blue seven. So let's do that. So here we can say x times x is going to be x squared. X times, we have a negative here, so we can say negative four is going to be negative four x. And just like, that we get x squared minus four x. And then over here we have seven times x so that's going to be plus seven x. And then we have seven times the negative four which is negative 28. And we are almost done. We can simplify it a little bit more. We have two first degree terms here. If I have negative four xs and to that I add seven xs, what is that going to be? Well those two terms together, these two terms together are going to be negative four plus seven xs. Negative four plus, plus seven. Negative four plus seven xs. So all I'm doing here, I'm making it very clear that I'm adding these two coefficients, and then we have all of the other terms. We have the x squared. X squared plus this and then we have, and then we have the minus, and then we have the minus 28. And we're at the home stretch! This would simply to x squared. Now negative four plus seven is three, so this is going to be plus three x. That's what these two middle terms simplify to, to three x. And then we have minus 28. Minus 28. And just like that, we are done! And a fun thing to think about, since it's in the same form. If we were to compare a is one, b is three, and c is -28, but it's interesting here to look at the pattern when we multiplied these two binomials. Especially these two binomials where the coefficient on the x term was a one. Notice we have x times x, that what actually forms the x squared term over here. We have negative four, let me do this in a new color. We have negative four times, that's not a new color. We have, we have negative four times seven, which is going to be negative 28. And then how did we get this middle term? How did we get this three x? Well, you had the negative four x plus the seven x. Or the negative four plus the seven times x. You had the negative four plus the seven, plus the seven times x. So I hope you see a little bit of a pattern here. If you're multiplying two binomials where the coefficients on the x term are both one. It's going to be x squared. And then the last term, the constant term, is going to be the product of these two constants. Negative four and seven. And then the first degree term right over here, it's coefficient is going to be the sum of these two constants, negative four and seven. Now this might, you could do this pattern if you practice it. It's just something that will help you multiply binomials a little bit faster. But it's super important that you realize where this came from. This came from nothing more than applying the distributive property twice.