1.14: Remainders and factorisation of a cubic

0606/13/M/J/20 — Question 5 · 8 marks

p(x)=6x3+ax2+12x+bp(x) = 6x^{3} + ax^{2} + 12x + b, where aa and bb are integers. p(x)p(x) has a remainder of 11 when divided by x3x - 3 and a remainder of 21-21 when divided by x+1x + 1.
(a)
Given that p(x)=(x2)Q(x)p(x) = (x - 2)Q(x), find Q(x)Q(x), a quadratic factor with numerical coefficients.
[6]
(b)
Hence solve p(x)=0p(x) = 0.
[2]