3.3: Factor and remainder theorems for a cubic

0606/11/M/J/21 — Question 3 · 8 marks

The polynomial

p(x) = ax^{3} - 9x^{2} + bx - 6

, where

a

and

b

are constants, has a factor of

x - 2

. The polynomial has a remainder of

66

when divided by

x - 3

(a)

Find the value of

a

and of

b

[4]

(b)

Using your values of

a

and

b

, show that

p(x) = (x - 2)q(x)

, where

q(x)

is a quadratic factor to be found.

[2]

(c)

Hence show that the equation

p(x) = 0

has only one real solution.

[2]