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CCSS.Math: , ,

we already have the tools in our toolkit to factor something like x squared plus 4x minus 5 and the way that we've already thought about it is we said hey let's think of two numbers that if we were to take their product we get negative 5 and if we were to add the two numbers we get positive 4 and there's a the fact that their product is negative tells you one of them is going to be positive and one of them is going to be negative and so there's a couple of ways you could think about it well you could say well maybe one of the numbers is negative one and then the other one is positive five actually this one seems to work negative 1 times 5 is negative 5 negative 1 plus 5 is positive 4 so this one actually seems to work the other option would have been since we're just going to deal with the the factors of 5 and 5s a prime number the other option would have been something like 1 and negative 5 there's only two factors for 5 so 1 and negative 5 the product would have been negative 5 but if you were to add these two numbers you would have gotten a negative 4 right over here so we're going to go with this right over here and so this tells us that if we want to factor this using kind of the tools that we already know about we will get and let me let me write these numbers in a different color so we can keep track of them so negative 1 & 5 we know that this would factor out to be X X minus 1 X minus 1 x times X plus 5 times X plus 5 and you can verify this for yourself that if you were to multiply this out you will get x squared plus 4x minus 5 you can even see this here x times X is x squared negative x plus 5 X is going to be 4x and then negative 1 times 5 is negative 5 fair enough this is all a review for us at this point now I want to talk tackle something a little bit a little bit more interesting let's say we wanted a factor let's say we wanted to factor x squared plus 4x y minus 5 y squared and at first this looks really daunting all of a sudden I've introduced a Y and a y squared here I have two variables how would I tack but the important thing is to just take a deep breath and realize that we're not fundamentally doing something different now the one little tricky thing I've done when I've written it this way and I encourage you to pause this and try this on your own before I explain any further but the one tricky thing I did right over here is I wrote the X before the Y and that tends to be the convention you just write them kind of in alphabetical order but if we want it in a form that's a little bit closer to this and something that we could that would fit this mold a little bit more is if we if we if we swap these two because then we could write it as x squared plus 4y x minus 5y squared and now it becomes pretty clear that this 4y term right over here this right over here is the coefficient on the x-term the same way that 4 was the coefficient on X right here and this negative 5y squared corresponds to the negative 5 right over here so we can do the exact same thought process let's think of 2 now not just numbers they're going to have variables at them let's think of two terms or two expressions that if I multiply them I get negative 5y squared and that when I take the sum I get 4y so let's think about how we could do this so one option one option would be positive let's say positive Y and positive Y and negative 5y so where would this take us positive Y times negative 5y would indeed be equal to negative 5y squared but then if I add Y to negative 5y I'm going to get negative 4y so this doesn't work let's see if we swap the two the two signs so what about negative Y and positive 5y well here if I take the product of negative Y and 5y it will be negative 5y squared and if I take the sum negative Y + 5 y it will be positive 4y so we know how to factor it now so once again let me put this in the same color so this is I'm going to put in this this move color this light purple and this I'll put it in a darker purple so now we know how to factor this this is going to be and this is the same exact mold that we did up here same exact idea this is going to be X instead of just a minus 1 here now we've factored here we factored into a negative 1 and 5 here we factor to a negative y + 5 y so instead of a negative 1 it's going to be a negative Y X minus y times X plus 5y X plus 5y and we can verify that when you multiply this out it indeed does equal x squared + 4 XY minus 5y so let me do that here just so we can know for sure so x times X is going to be x squared X let me do everything in a different color x times 5y is going to be plus 5 X Y then negative Y times X is negative Y X and then finally finally if we take and I'm running out of colors if we take negative Y times 5y that's negative 5y squared and now we just have to simplify we have to combine these middle two terms right over here and at first it looks a little bitter this is X Y this is YX not so obvious but we just have to rewrite it this is the same thing as 5 YX minus y X and so here you're saying look I have 5y X's and I'm going to subtract Y X's so I'm gonna have 4y X's so this is just going to be 4y X's I have 5y X's take out another YX I'm going to have 4y X's so this is going to be x squared plus 4y x minus 5y squared and it all works out