Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2015 / Paper 1 / 1

Group the polynomial into two pairs, then use the difference of two squares:Thus $1$ is a double polynomial root and $-1$ is a simple polynomial root.

The repeated factor requires both a simple-pole and a double-pole term in the partial fraction decomposition. WriteMultiplying through gives $1=A(x-1{)}^2+B(x^2-1)+C(x+1)$. Substituting $x=-1$ and $x=1$ gives $A=1/4$ and $C=1/2$; the quadratic coefficient then gives $B=-1/4$. Therefore