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?