1. ## [SOLVED] Polynomial division

Not really a homework question, but I have a test tomorrow and I need to be able to solve these kind of problems.

Find the remainder when the polynomial $\displaystyle x^{100} + x^{67} - x^{32} - 2x^9 +1$ is divided with:

a) $\displaystyle x - 1$
b) $\displaystyle x + 2$
c) $\displaystyle x^2 - x$

I already have the answers, what I'm looking for is how to solve the problems. Polynomial long division would easily fill up several pages, so I'm clueless as what to do.

2. Originally Posted by Spec
Not really a homework question, but I have a test tomorrow and I need to be able to solve these kind of problems.

Find the remainder when the polynomial $\displaystyle x^{100} + x^{67} - x^{32} - 2x^9 +1$ is divided with:

a) $\displaystyle x - 1$
b) $\displaystyle x + 2$
c) $\displaystyle x^2 - x$
!
Theorem: Given $\displaystyle f(x),g(x)$ two polynomials over $\displaystyle \mathbb{Z}$ with $\displaystyle g(x)\not = 0$ then $\displaystyle f(x) =q(x)g(x)+r(x)$ and $\displaystyle \deg r(x) < \deg g(x) \mbox{ or }r(x)=0$.

So,
$\displaystyle x^{100} + x^{67} - x^{32} - 2x^9 +1=q(x)(x-1)+k$.
It is a constant since the degree is zero.
Since the above statement is true for all $\displaystyle x$ let $\displaystyle x=1$ to get:
$\displaystyle 0=q(1)(0)+k\implies k=0$.
Thus there is no remainder.

Try the same approach to the other ones.