2025 IGCSE Additional Mathematics May/June 0606/12 Q5

The polynomial \(\mathrm p\) is such that \(\mathrm p(x)=3x^3-7x^2+ax+b\), where \(a\) and \(b\) are integers.

It is given that \(\mathrm p'(-1)=21\) and that \(x-2\) is a factor of \(\mathrm p(x)\).

1. Find the values of \(a\) and \(b\).[4]
2. Hence write \(\mathrm p(x)\) as a product of linear factors with integer coefficients.[3]
3. Using your values of \(a\) and \(b\), solve the equation \(3\mathrm e^{6y}-7\mathrm e^{4y}+a\mathrm e^{2y}+b=0\).[3]