0606 P13 - Nov 2020 - Q7 - 8 marks
8191
The polynomial
\(\mathrm{p}(x)=ax^3+bx^2-19x+4,\)
where \(a\) and \(b\) are constants, has a factor \(x+4\) and is such that
\(2\mathrm{p}(1)=5\mathrm{p}(0).\)
(a) Show that
\(\mathrm{p}(x)=(x+4)(Ax^2+Bx+C),\)
where \(A\), \(B\) and \(C\) are integers to be found.
(b) Hence factorise \(\mathrm{p}(x)\).
(c) Find the remainder when \(\mathrm{p}'(x)\) is divided by \(x\).
