3.4: Cubic with second derivative, factor and linear factors

0606/11/M/J/23 — Question 2 · 9 marks

The polynomial

p

is such that

p(x) = ax^{3} + 7x^{2} + bx + c

, where

a

,

b

and

c

are integers.

(a)

Given that

p''\!\left(\dfrac{1}{2}\right) = 32

, show that

a = 6

.

[2]

(b)

Given that

p(x)

has a factor of

3x - 4

and a remainder of

7

when divided by

x + 1

, find the values of

b

and

c

.

[4]

(c)

Write

p(x)

in the form

(3x - 4)q(x)

, where

q(x)

is a quadratic factor.

[2]

(d)

Hence write

p(x)

as a product of linear factors with integer coefficients.

[1]