2026 IGCSE Additional Mathematics May/June 0606/11 Q5

The polynomial \(\mathrm{p}\) is such that \(\mathrm{p}(x)=ax^3+4x^2-ax-28\), where \(a\) is a positive constant.

It is given that \(\mathrm{p}(a)=0\).

1. Find the value of \(a\).[4]
2. Using your value of \(a\), find \(\mathrm{p}(x)\) as a product of a linear factor and a quadratic factor.[2]
3. Hence show that the equation \(\mathrm{p}(x)=0\) has exactly one root.[2]