1.10: Constants from divisibility and derivative conditions

0606/12/O/N/18 — Question 9 · 8 marks

The polynomial p(x)=ax3+bx2+cx−9p(x) = ax^{3} + bx^{2} + cx - 9 is divisible by x+3x + 3. It is given that p′(0)=36p'(0) = 36 and p′′(0)=86p''(0) = 86.
(i)
Find the value of each of the constants aa, bb and cc.
[6]
(ii)
Using your values of aa, bb and cc, find the remainder when p(x)p(x) is divided by 2x−12x - 1.
[2]