Polynomial special products: difference of squares

earlier in our mathematical adventures we had expanded things like X plus y times X minus y just as a bit of review this is going to be equal to x times X which is x squared plus x times negative Y which is negative XY plus y times X which is plus XY and then minus y times y or you could say Y times the negative Y so it's going to be minus y squared negative XY positive XY so this is just going to simplify to x squared minus y squared and this is all a review we covered it and when we thought about factoring things that are differences of squares we thought about this when we were first learning to multiply binomials and what we're going to do now is essentially just do the same thing but do it with slightly more complicated expressions and so another way of expressing what we just did is we could also write something like a plus B times a minus B is going to be equal to what well it's going to be equal to a squared minus B squared the only difference between what I did up here and what I did over here is instead of an X I wrote an a instead of a why I wrote a B so given that let's see if we can expand and then combine like terms for if I'm multiplying these two expressions say I'm multiplying 3 plus 5x to the 4th times 3 minus 5x to the fourth pause this video and see if you can work this out all right well there's two ways to approach it you could just approach it exactly the way that I approached it up here but we already know that when we have this pattern where we have something plus something times that same original something minus the other or something well that's going to be of the form of this thing squared minus this thing squared and remember the only reason why I'm applying that is I have a 3 right over here and here so the 3 is playing the role of the a so let me write that down that is our a and then the role of the B is being played by 5 X to the 4th so that is our B right over there so this is going to be equal to a squared minus B squared but our a is 3 so it's going to be equal to 3 squared minus and then our B is 5 X to the fourth minus 5x to the fourth squared now what does all of this simplify to well this is going to be equal to 3 squared is 9 and then minus 5 X to the fourth squared let's see 5 squared is 25 and then X to the 4th squared well that is just going to be X to the 4 times X to the fourth which is just X to the 8th another way to think about is our exponent properties we are this is same thing as five squared times X to the fourth squared if I raise something exponent and then raise that to another exponent I multiply the exponents and there you have it let's do another example let's say that I were to ask you what is 3y squared plus 2y to the fifth times 3y squared minus 2y to the fifth pause this video and see if you can work that out well we're going to do it the same way you can of course always just try to expand it out the way we did originally but we could recognize here that hey I have an A plus a B times the a minus a B so that's going to be equal to our a squared so what's 3y squared well that's going to be 9 Y to the fourth minus our B squared well what's 2 Y to the fifth squared well 2 squared is 4 and Y to the fifth squared is y to the 5 times 2y to the 10th power and there's no further simplification that I could do here I can't combine any like terms and so we are done here as well