Difficult polynomial question involving factor and remainder theorems

[stfz]

Homework Statement

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

Homework 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.

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
I could do most of the other questions in the exercise, but not this one (and other related ones!)

Any help would be appreciated

 

[SammyS]

Homework Statement

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

Homework 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.

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
I could do most of the other questions in the exercise, but not this one (and other related ones!)

Any help would be appreciated

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

 

[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?

 

[Mark44]

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?

http://en.wikipedia.org/wiki/Polynomial_long_division

 

[SteamKing]

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?

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.

 

[stfz]

Woops, got what I missed. Thanks!

 

[epenguin]

It (one of the most useful things to remember) came up here just yesterday:

https://www.physicsforums.com/showpost.php?p=4704351&postcount=1
https://www.physicsforums.com/showpost.php?p=4704833&postcount=3