1.15: Polynomial divisible by a linear factor with trig application

0606/13/O/N/18 — Question 11 · 10 marks

The polynomial

p(x) = ax^{3} + 17x^{2} + bx - 8

is divisible by

2x - 1

and has a remainder of

-35

when divided by

x + 3

.

(i)

By finding the value of each of the constants

a

and

b

, verify that

a = b

.

[4]

(ii)

Using your values of

a

and

b

, find

p(x)

in the form

(2x - 1)q(x)

, where

q(x)

is a quadratic expression,

[2]

(iii)

factorise

p(x)

completely,

[1]

(iv)

solve

a \sin^{3} \theta + 17 \sin^{2} \theta + b \sin \theta - 8 = 0

for

0^{\circ} < \theta < 180^{\circ}

.

[3]