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Thread: Greatest common divisor (different to last thread)

  1. #1
    Senior Member chella182's Avatar
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    Greatest common divisor (different to last thread)

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

    Quite literally stumped. I get why $\displaystyle d(X)$ is of degree 1, but I don't know how to determine it at all.
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  2. #2
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    Quote Originally Posted by chella182 View Post

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

    Quite literally stumped. I get why $\displaystyle d(X)$ is of degree 1, but I don't know how to determine it at all.
    well, it's pretty obvious that $\displaystyle d(x)=x+2$ because $\displaystyle d(x)$ must divide $\displaystyle f(x) + x^2+x+1 - f(x) = x^2+x+1=(x+2)^2.$
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