3.29: Polynomial coefficients from remainder data

0606/11/O/N/23 — Question 2 · 8 marks

The polynomial P(x)=ax3−11x2+bx+cP(x) = ax^{3} - 11x^{2} + bx + c is divisible by xx, has remainder 32\dfrac{3}{2} when divided by 2x+12x + 1, and P′(2)=18P'(2) = 18.
(a)
Find the values of aa, bb and cc.
[6]
(b)
Hence write P(x)P(x) as a product of three linear factors.
[2]