Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2025 / Paper 1 / 8 / Solution

For the partial fraction decomposition, multiply by $(x-1)(x-2)$ to obtain the polynomial identityComparing the $x^2$ coefficients gives $a=1$. Substitution of $x=1$ gives $2=-b$, while $x=2$ gives $5=c$. ThereforeThe original identity holds for $x\ne 1,2$; evaluating the multiplied polynomial identity at those values is valid.