3.6: Factor and derivative conditions; full factorisation

0606/23/M/J/22 — Question 6 · 8 marks

Do not use a calculator in part (b).

The polynomial p(x)p(x) is such that p(x)=6x3+ax2−52x+bp(x) = 6x^{3} + ax^{2} - 52x + b, where aa and bb are integers. It is given that p(x)p(x) is divisible by 2x−32x - 3 and that p′(1)=4p'(1) = 4.
(a)
Find the values of aa and bb.
[5]
(b)
Using your values of aa and bb, factorise p(x)p(x) fully.
[3]