Video Transcript: Factoring Polynomials with Common Factors

Intro to factors & divisibility 

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.

Factoring with the distributive property 

What I want to do is start with an expression like 4x plus 18 and see if we can rewrite this as the  product of two expressions. Essentially, we're going to try to factor this. And the key here is to  figure out are there any common factors to both 4x and 18? And we can factor that common  factor out. We're essentially going to be reversing the distributive property. So for example,  what is the largest number that is-- or I could really say the largest expression-- that is divisible  into both 4x and 18? Well, 4x is divisible by 2, because we know that 4 is divisible by 2. And 18 is  also divisible by 2, so we can rewrite 4x as being 2 times 2x. If you multiply that side, it's  obviously going to be 4x. And then, we can write 18 as the same thing as 2 times 9. And now it  might be clear that when you apply the distributive property, you'll usually end up with a step  that looks something like this. Now we're just going to undistribute the two right over here.  We're going to factor the two out. Let me actually just draw that. So we're going to factor the  two out, and so this is going to be 2 times 2x plus 9. And if you were to-- wanted to multiply this out, it would be 2 times 2x plus 2 times 9. It would be exactly this, which you would simplify as  this, right up here. So there we have it. We have written this as the product of two expressions,  2 times 2x plus 9. Let's do this again. So let's say that I have 12 plus-- let me think of something  interesting-- 32x. Actually since we-- just to get a little bit of variety here, let's put a y here, 12  plus 32y. Well, what's the largest number that's divisible into both 12 and 32? 2 is clearly  divisible into both, but so is 4. And let's see. It doesn't look like anything larger than 4 is  divisible into both 12 and 32. The greatest common factor of 12 and 32 is 4, and y is only  divisible into the second term, not into this first term right over here. So it looks like 4 is the  greatest common factor. So we could rewrite each of these as a product of 4 and something  else. So for example, 12, we can rewrite as 4 times 3. And 32, we can rewrite-- since it's going to  be plus-- 4 times. Well if you divide 32y by 4, it's going to be 8y. And now once again, we can  factor out the 4. So this is going to be 4 times 3 plus 8y. And once you do more and more  examples of this, you're going to find that you can just do this stuff all at once. You can say hey,  what's the largest number that's divisible into both of these? Well, it's 4, so let me factor a 4  out. 12 divided by 4 is 3. 32y divided by 4 is 8y.