3.39: Polynomial factors and second-derivative test

0606/23/M/J/24 — Question 6 · 11 marks

Do not use a calculator in this question.

(a)

Given that

x - 3

and

x + 1

are both factors of

2x^{3} - 3x^{2} - 8x - 3

, solve

2x^{3} - 3x^{2} - 8x - 3 = 0

[2]

(b)(i)

The polynomial

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

has remainder

-5

when divided by

x - 1

. The curve

y = p(x)

has stationary points at

x = \dfrac{3}{4}

and

x = 2

. Find

a

b

and

c

[7]

(b)(ii)

Hence use the second derivative test to show that the stationary point at

x = 2

is a minimum.

[2]