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Determine two polynomials $h(x), k(x) \in \mathbb Q[x]$ having rispectively $1,-1,2$ and $1,2,-2$ as roots. Explain why $t(x) = x-1$ is a divisor for every $\gcd(h(x),k(x))$.

I figured $h(x) =(x-1)(x+1)(x-2)=x^3-2x^2-x+2$ and $k(x)= (x-1)(x-2)(x+2)= x^3-x^2-4x+4$, but I just can't find a way to explain wht $t(x)$ is a divisor, can anyone please help me out and give me a hint?

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From the way you constructed $h$ and $k$, $x-1$ occurs as a factor for both. By unique factorisation, the gcd is in fact $(x-1)(x-2)$.

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    is that all? I was looking for some complicated theorem and all I need to do was looking under my nose? :P2012-08-29