Target Mathematics

1.11: Polynomial remainders, polynomial division and polynomial roots

4PM1/2R/November/2020 — Question 2 · 11 marks

f(x)=x3+px+q,f(x)=x^3+px+q,
where pp and qq are constants. The remainder when f(x)f(x) is divided by (x−1)(x-1) is −12-12. The remainder when f(x)f(x) is divided by (x−4)(x-4) is 3030.
(a) Find the value of pp and the value of qq.
(6)
Using your values of pp and qq,
(b) show that f(3)=0f(3)=0,
(1)
(c) express f(x)f(x) as a product of linear factors,
(3)
(d) hence solve f(x)=0f(x)=0.
(1)