# [SOLVED] Polynomial division

• Aug 28th 2007, 12:21 PM
Spec
[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 $x^{100} + x^{67} - x^{32} - 2x^9 +1$ is divided with:

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

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.

• Aug 28th 2007, 01:40 PM
ThePerfectHacker
Quote:

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 $x^{100} + x^{67} - x^{32} - 2x^9 +1$ is divided with:

a) $x - 1$
b) $x + 2$
c) $x^2 - x$
!

Theorem: Given $f(x),g(x)$ two polynomials over $\mathbb{Z}$ with $g(x)\not = 0$ then $f(x) =q(x)g(x)+r(x)$ and $\deg r(x) < \deg g(x) \mbox{ or }r(x)=0$.

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

Try the same approach to the other ones.