Greatest common divisor (different to last thread)

**chella182** · April 18th 2010, 04:20 PM

Let $f(X)$ be a polynomial over $\mathbb{Z}_3$ such that $d(X)=gcd(f(X),f(X)+(X^2+X+1))$ is a polynomial of degree 1. Determine $d(X)$ .

Quite literally stumped. I get why $d(X)$ is of degree 1, but I don't know how to determine it at all.

**NonCommAlg** · April 19th 2010, 06:45 AM

Originally Posted by chella182

Let $f(X)$ be a polynomial over $\mathbb{Z}_3$ such that $d(X)=gcd(f(X),f(X)+(X^2+X+1))$ is a polynomial of degree 1. Determine $d(X)$ .

Quite literally stumped. I get why $d(X)$ is of degree 1, but I don't know how to determine it at all.

well, it's pretty obvious that $d(x)=x+2$ because $d(x)$ must divide $f(x) + x^2+x+1 - f(x) = x^2+x+1=(x+2)^2.$

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