Some questions about math factor proves

[furtivefelon]

hi, i have three math factor proves i can't figure out:

1. Show that x-y is a factor of x^n-y^n
Through empircal testing, i figured out the other factor must be x^(n-1) + x^(n-2)y + x^(n-3)y^2 + ... + y^(n-1), though i can't come up with a prove for it..

2. Prove that (x+a) is a factor of (x+a)^5 + (x+c)^5 + (a-c)^5

3. prove that (x-a) is a factor of x^3 - (a+b+c)x^2 + (ab+bc+ca)x - abc

the last two i have no idea where to begin.. Can someone please help me? thanks :D

 

[HallsofIvy]

Try this: (x- a) is a factor of polynomial p(x) if and only if p(a)= 0.

What happens if you put x= a in each of those polynomials?

 

[Hurkyl]

x^(n-1) + x^(n-2)y + x^(n-3)y^2 + ... + y^(n-1), though i can't come up with a prove for it..

Well, what do you get when you multiply this by (x - y)? If you get x^n - y^n, then you have a proof! (not a 'prove')