# Homework Help: Help proving polynomial identity

1. Apr 12, 2010

### alyks

1. The problem statement, all variables and given/known data

Prove the following when p is a positive integer:
$$b^p - a^p = (b-a)(b^{p-1}+b^{p-2}a+b^{p-3}a^2+...+ba^{p-2}+a^{p-1})$$

Hint: Use the telescoping property for sums.

2. Relevant equations
None

3. The attempt at a solution

I tried reducing it to, $$(b-a)\sum_{k=1}^p b^{p-k}a^{k-1}$$ but I wasn't able to do anything with it.

I've been trying to work on this exercise which is a part of a problem set in induction. But I've been having quite a bit of difficulty and I'd appreciate it if somebody here could give me a hint as to the general direction. It's been bugging me for the past few days.

2. Apr 12, 2010

### epenguin

I don't know what telescoping is but er

$$(b-a)(b^{p-1}+b^{p-2}a+b^{p-3}a^2+...+ba^{p-2}+a^{p-1})$$

is $$b(b^{p-1}+b^{p-2}a+b^{p-3}a^2+...+ba^{p-2}+a^{p-1}) -a(b^{p-1}+b^{p-2}a+b^{p-3}a^2+...+ba^{p-2}+a^{p-1})$$.

If you can't see what that gives, expand it out by executing the multiplications and collecting up terms.

Quite a useful and important formula. When you do geometric series the same one is involved - make the relation.