2025 IGCSE Additional Mathematics February/March 0606/12 Q4

The polynomial \(\mathrm{p}\) is given by \(\mathrm{p}(x)=a^2x^3+2ax^2+ax+2\), where \(a\) is a positive integer.

It is given that \(2x+1\) is a factor of \(\mathrm{p}(x)\).

1. Find the value of \(a\).[3]
2. Hence factorise \(\mathrm{p}(x)\).[2]
3. Hence show that the equation \(\mathrm{p}(x)=0\) has only one real root.[1]