2024 IGCSE Additional Mathematics October/November 0606/13 Q5

The polynomial \(\mathrm{p}\) is such that \(\mathrm{p}(x)=ax^3+bx^2-19x+c\), where \(a\), \(b\) and \(c\) are integers. It is given that \(x+2\) is a factor of \(\mathrm{p}(x)\). When \(\mathrm{p}(x)\) is divided by \(x+1\) the remainder is \(20\).

1. Show that \(7a-3b=39\).[3]

It is also given that when \(\mathrm{p}'(x)\) is divided by \(x-1\) the remainder is \(1\).

2. Find the values of \(a\), \(b\) and \(c\).[3]