1. ## Use identity to prove result about polynomials

Hi there,

I am stuck on the following problem:

If f(x) is a polynomial and a is a number, prove that there exists a polynomial g(x) such that f(x) = (x - a)g(x) + f(a)

Hint: use the identity $\displaystyle x^n -a^n = (x - a)(x^{n-1} + ax^{n-2} + ... + a^{n-2}x + a^{n-1})$

I can't really see how to apply the hint. Apart from the obvious inclusion of (x - a), I'm out of ideas.

Many thanks!

2. ## A further hint

Let $\displaystyle f(x)=b_nx^n+b_{n-1}x^{n-1}+\ldots+b_2x^2+b_1x+b_0$. Now look at $\displaystyle f(x)-f(a)$.

--Kevin C.

3. Aha, gotcha! Very nice indeed. Thankyou!