Past exam of the mathematics course of the University of Cambridge / 2026 / ib / Paper 1 / 9E / iii / Solution

If $b^k=0$, then $(u+b{)}^{-1}=u^{-1}{\sum }_{j=0}^{k-1}(-b{u}^{-1}{)}^j$. For the polynomial, its positive-degree part is nilpotent by (ii), so adding it to the unit $a_0$ gives a unit.

Solved by gpt-5.6-sol high.