# Find GCD of functions/polynomials [duplicate]

Let $\mathbb F$ be any field $a \neq b$ be two elements of $\mathbb F$

Find the GCD of $f(x) = x + a$ and $g(x) = x + b$. Also find the polynomials $s(x)$ and $t(x)$ such that $s(x)f(x) + t(x)g(x)$ equals the GCD.

My work so far:

$d(x) = s(x)f(x)+t(x)g(x)$

$d(x) = s(x)[x+a] + t(x)[x+b]$

$d(x) = x[s(x) + t(x)] + s(x)*a +t(x)*a$

Thoughts on how I can find the polynomials? Is there an explicit solution for the GCD $d(x)$?

## marked as duplicate by Tom Oldfield, Andreas Caranti, Ross Millikan, user23500, vonbrandApr 3 '13 at 21:44

• Now that you know that, you could try to prove that the gcd of $X^n-1$ and $X^m-1$ is $X^k-1$ with $k$ the gcd of $m$ and $n$. – Julien Apr 3 '13 at 21:54
if $a\neq b$, theyre coprime and $[(x+a)-(x+b)](a-b)^{-1}=1$