Polynomials, equivalent equations

[mstcrow5429]

Why doesn't (x - y)^m = x^m - y^m?

Using the rule that (ab)^n = a^n b^n, I thought that the first equation would be true. I've plugged in some integers, and it's not true, but I don't know why. Thanks.

[Blue]

i don't know the "textbook reason" but from what i know about polynomials, your equation doesn't equal because (x - y) is different from (ab) so you can't use that rule. for instance, try substituting the variable "m" with a number (let's say 2). so it would be:

(assume m=2)
(x - y)^2 = (x - y)(x - y)
if you calculated it, it would come out to be x^2 - 2xy + y^2
so if you switch back to using the variable, it would be:
x^m - mxy + y^m

do you kinda see how that equation works? i'm sorry i don't know a textbook answer =(