0606 P13 - Nov 2022 - Q4 - 7 marks
7875
The polynomial \(p(x)\) is such that
\(p(x)=ax^3+13x^2+bx+c,\)
where \(a\), \(b\) and \(c\) are integers. It is given that \(p'(0)=-9\).
(a) Show that \(b=-9\).
It is also given that \(3x+2\) is a factor of \(p(x)\) and that when \(p(x)\) is divided by \(x+1\) the remainder is \(6\).
(b) Find the values of \(a\) and \(c\).
(c) Find the quadratic \(q(x)\) such that
\(p(x)=(3x+2)q(x).\)
(d) Hence find \(p(x)\) as a product of linear factors with integer coefficients.
