Sal explains what it means for a polynomial to be a factor of another polynomial, and what it means for a polynomial to be divisible by another polynomial.
- [Voiceover] You're probably familiar with the general term factor. So if I were to say: What are the factors of 12, you could say: Well what are the whole numbers that I can multiply by another whole number to get 12? So for examples, you could say things like, well I could multiply one times 12 to get 12. So you could say that one is a factor of 12. You could even say that 12 is a factor of 12. You could say two times six is equal to 12, so you could say that two is a factor of 12 and that six is also a factor of 12. And of course three times four is also 12, so both three and four are factors of 12. So if you said well what are the factors of 12 and you've seen this before, well you could say: one, two, three, four, six, and 12, those are all factors of 12. And you could also phase it the other way around, so let me just give an example. So if I were to pick on three, I could say that three is a factor of 12. Or to phrase it slightly differently, I could say that 12 is divisible, 12 is divisible by three. Now what I wanna do in this video is extend this idea of being a factor or divisibility into the algebraic world. So for example, if I were to take 3xy. So this is a monomial with an integer coefficient. Three is an integer right over here. And if I were to multiply it with another monomial with an integer coefficient, I don't know, let's say times negative two X squared, Y to the third power, what is this going to be equal to? Well this would be equal to, if we multiply the coefficients, three times negative two is going to be negative six. X times X squared is X to the third power. And then Y times Y to the third is Y to the fourth power. And so what we could say is, if we wanted to say factors of negative six X to the third, Y to the fourth, we could say that 3xy is a factor of this just as an example; so let me write that down. We could write that 3xy is a factor of, is a factor of... of negative six X to the third power, Y to the fourth, or we could phrase that the other way around. We could say that negative six X to the third, Y to the fourth, is divisible by, is divisible by 3xy. So hopefully you're seeing the parallels. If I'm taking these two monomials with integer coefficients and I multiply 'em and I get this other, in this case, this other monomial, I could say that either one of these and there's actually other factors of this, but I could say either one of these is a factor of this monomial, or we could say that negative six X to the third, Y to the four is divisible by one of its factors. And we could even extend this to binomials or polynomials. For example, if I were to take, if I were to take, let me scroll down a little bit, whoops, if I were to take, let me say X plus three and I wanted to multiply it times X plus seven, we know that this is going to be equal to, if I were to write it as a trinomial, it's gonna be X times X, so X squared, and then it's gonna be three X plus seven X, so plus 10x; and if any of this looks familiar, we have a lot of videos where we go in detail of multiplying binomials like this. And then three times seven is 21. Plus 21. So because I multiplied these two, in this case binomials, or we could consider themselves to be polynomials, polynomials or binomials with integer coefficients. Notice the coefficients here, they're one, one. The constants here, they're all integers. Because I'm dealing with all integers here, we could say that either one of these binomials is a factor of this trinomial, or we could say this trinomial is divisible by either one of these. So let me write that down. So I could say, I'll just pick on X plus seven. We could say that X plus seven is a factor, is a factor of X squared plus 10x plus 21; or we could say that X squared plus 10x plus 21 is divisible by, is divisible by I could say X plus three or I could say X plus seven is divisible by, X plus seven. And the key is, is that both of these binomials, or even if we were dealing with polynomials, we are dealing with things that have integer, we're dealing with things that have integer coefficients.