Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics IV / 7 / a / ii / Solution

Let $P=3x^4-5x^3+2x^2-3x+2$ and $Q=3x^3+x^2+x-2$. Applying polynomial long division gives the euclidean algorithmThus their polynomial greatest common divisor is $3x-2$, up to a nonzero constant factor. Their unique common root of a polynomial is therefore