Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2012 / Paper 1 / 3 / Solution

By the polynomial remainder theorem, the first polynomial remainder is $f(1)=23-2p$. Equating it to $9$ gives $p=7$.

For the separate polynomial long division, successively cancel the leading powers of the dividend. Subtracting $x^2(x-1)$ leaves $2x^2+2x+1$; subtracting $2x(x-1)$ leaves $4x+1$; subtracting $4(x-1)$ leaves $5$. ThusThe polynomial remainder also equals the dividend evaluated at $x=1$.