1.12: Divisibility conditions on a cubic and its derivative

0606/13/M/J/16 — Question 10 · 8 marks

(i)

Given that

f(x) = 4x^{3} + kx + p

is exactly divisible by

x + 2

and

f'(x)

is exactly divisible by

2x - 1

, find the value of

k

and of

p

[4]

(ii)

Using the values of

k

and

p

found in part (i), show that

f(x) = (x + 2)(ax^{2} + bx + c)

, where

a

b

and

c

are integers to be found.

[2]

(iii)

Hence show that

f(x) = 0

has only one solution and state this solution.

[2]