2.12: Factor and remainder on a cubic

0606/23/O/N/16 — Question 4 · 8 marks

The cubic given by p(x)=x3+ax2+bx−24p(x) = x^{3} + ax^{2} + bx - 24 is divisible by x−2x - 2. When p(x)p(x) is divided by x−1x - 1 the remainder is −20-20.
(i)
Form a pair of equations in aa and bb and solve them to find the value of aa and of bb.
[4]
(ii)
Factorise p(x)p(x) completely and hence solve p(x)=0p(x) = 0.
[4]