# Homework Help: Difficult polynomial question involving factor and remainder theorems

1. Mar 31, 2014

### stfz

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

Prove that $(a-b)$ is a factor of $a^5-b^5$, and find the other factor.

2. Relevant equations

Remainder theorem : remainder polynomial $p(x)$ divided by $(x-a)$ is equal to $p(a)$
Factor theorem : if remainder = 0, then divisor was a factor of dividend.

3. The attempt at a solution

I think am able to prove that it is a factor:

$P(x) = x^5 - b^5$; we replace a with x
$P(x) = (x-b)Q(x) + 0$ ; we assume that (x-b) is a factor
$P(b) = (b-b)Q(x) + 0 = 0$; proves that .. um.. I think I'm going the wrong way anyway. This doesn't really prove anything? er.. eh.. ?

And I can't really find the other factor :grumpy:
I could do most of the other questions in the exercise, but not this one (and other related ones!)

Any help would be appreciated

Last edited: Mar 31, 2014
2. Mar 31, 2014

### SammyS

Staff Emeritus
Do you know long division with polynomials, or synthetic division ?

Last edited: Mar 31, 2014
3. Mar 31, 2014

### stfz

Yes. However, attempting polynomial division on (x^5+b^5) didn't quite work for me. If that's how to solve the question, could someone give me an example?

4. Mar 31, 2014

5. Apr 1, 2014

### SteamKing

Staff Emeritus
Maybe that's because you want to divide (a$^{5}$-b$^{5}$) by (a - b).

Start off with a few warm-up exercises:

(a$^{2}$-b$^{2}$)/(a-b)

(a$^{3}$-b$^{3}$)/(a-b)

You should know the answer to the first exercise by inspection.

6. Apr 1, 2014

### stfz

Woops, got what I missed. Thanks!

7. Apr 1, 2014