Sal analyzes the factorization of 3y^3-100y as 4y(My+g)(My-g) to find the possible values for the missing coefficient g.
- [Voiceover] The polynomial expression, 36 y to the third minus 100y can be factored as 4y times all of this, My plus g, times My minus g, where M and g are integers. Sally wrote that g could be equal to 3. Brandon wrote that g could be equal to 10. Which student is correct? Now, when you look at this, it seems really daunting, all this M's and g's here, but we just need to realize that they're factoring out, first they factored out a 4y from the 36y to the third minus 100y, and it looks like whatever's left is a difference of squares, which they then factor even further. So I encourage you to pause the video and just factor this out as much as you can. First factoring out a 4y, and then we can think about what g is going to be equal to, or whether Sally or Brandon is correct. So, then now let's work through this together. So, if we look at, if we look at this expression right over here, we want to factor out a 4y, so 36y to the third minus 100y, that's the same thing as, 36y to the third is the same thing as 4y times, let's see, 4y times 9y squared, right? Because 4 times 9 is 36, and y times y squared is y to the third, so all I did to get the 9y squared is I divided 36 by 4 to get the 9, and I divided y to the third by the y to get y squared, so if you factor out a 4y, you're left with 9y squared for that first term, and then, for the second term, let's see, if we, we're going to subtract, if we factor out a 4y again, what's left over? 100 divided by 4 is 25, and then y divided by y is just 1, so we're left with a 25 here. So just to be clear what's going on, this 36y to the third, I just rewrote it as 4y times 9y squared. One way to think about it is I wrote it with the 4y factored out, and then, the 100y, right over here, I wrote it with the 4y factored out, so it's 4y times 25, and now it's very clear that we can factor out 4y from this entire thing, so we can factor out, you can think of it as undistributing the 4y, so this is going to be equal to 4y and what is left over? Well, if you factor out a 4y out of this first term, you're going to have a 9y squared, 9y squared, and then minus 25, and then we're going to be left with minus 25, and when we write it like this, we see what we have in parentheses here. This is a difference of squares, and, we could skip a step, but let me just rewrite it, so we could rewrite it as, literally, a difference of squares. 9y squared, that is the same thing as, that is the same thing as 3y, that whole thing to the 2nd power, 3 squared is 9, y squared is y squared, and then we have minus 25, we can rewrite as 5 squared, so you see, we have a difference of squares, and we've seen this pattern multiple times. If this is the first time you're seeing it, I encourage you to watch the videos on Khan Academy on difference of squares, but we know anything of the form, anything of the form a squared minus b squared, let me do it in that color, minus b squared can be factored as being equal to, this is equal to, if I were to write it as a product of two binomials, this is going to be equal to a plus b times a minus b, and you can verify that that works, if you've never seen this before, or you can watch those videos for review. So this right over here can be rewritten as 4y, which we factored out at the beginning, it's going to be times the product of two binomials for this part right over here, and so in this case, a is 3y, so it's going to be 3y plus 5, times 3y minus 5, so let me write that down. So 3y plus 5 times 3y minus 5. 3y minus 5. So now that we factored this, let's go back to what they originally told us, so then we have 4y, so this 4y corresponds to that 4y, right over there, and then you have My plus g, and then you have My minus g, so you could view the My, the My's right over there, that's the 3y right over there, so we could say that M is equal to 3, M is equal to 3, and then we do plus 5 and minus 5, plus g and minus g, so, g, if we're pattern matching right over here, g is going to be equal to 5. So g is equal to 5. So what's interesting about this problem is that neither one of them are correct, so, I could write, neither is correct. G is equal to 5. That was a tricky one.