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CCSS.Math: , ,

the polynomial expression 36 Y to the third minus 100 y can be factored as 4y times all of this my plus G times my minus G or M and G are integers Sally wrote the G could be equal to 3 Brandon rode the 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 factor out a 4y from the 36 Y to the 3rd minus 100 y 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 could think about what G is going to be equal to or whether Sally or Brandon is correct so now let's work through this let's work through this together so if we look at if we look at this expression right over here and we want to factor out a 4 y so 36 Y to the third minus 100 Y that's the same thing as 36 Y 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 that divided 36 by 4 to get the 9 and I divided Y to the 3rd by the Y to get Y squared so if you factor out a 4 Y you're left with 9y squared for that first term and then for this second term let's see if we we're going to subtract we factor out a 4 y again we factor out a 4 y what's left over 100 divided by 4 is 25 25 and then Y and then Y divided by y is just 1 so we're just left with a 25-year so just to be clear what's going on this 36 Y to the 3rd I just rewrote it as 4y times 9y squared one way to think about it is I wrote it with the 4y factor 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 could think of it as undistributing the 4y so this is going to be equal to this is going to be equal to 4y and what is left over well if you factor out a 4y of this first term we'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 second power 3 squared is 9 Y squared is y squared and then we have 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 and 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 minus 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 binome of two binomials this is going to be equal to a a plus B a plus B times a minus 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 re-written as for Y 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 3 y plus 5 times 3y minus 5 so let me write that down so 3y plus 5 plus 5 times 3y minus 5 3y minus 5 3y minus 5 so now that we factor this let's let's go back to what what they originally told us so that we have for y 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 is 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 G if we're pattern matching right over here G is going to be equal to 5 so G is equal to 5 so it's interesting about this problem is that neither one of them neither one of them are correct so I could write neither is correct G is equal to G is equal to 5 that was a tricky one