3.10: Polynomial factors then cosec equation

0606/12/F/M/24 — Question 5 · 13 marks

The polynomial

p

is such that

p(x) = 5x^{3} + ax^{2} + 39x + b

, where

a

and

b

are constants.

(a)

Given that

x + 3

is a factor of both

p(x)

and

p'(x)

, find the values of

a

and

b

.

[5]

(b)

Hence solve the equation

p(x) = 0

. You must show your working.

[3]

(c)

Hence, using your values for

a

and

b

, solve the equation

5\,\mathrm{cosec}^{3}\theta + a\,\mathrm{cosec}^{2}\theta + 39\,\mathrm{cosec}\,\theta + b = 0

for

0^{\circ} \leqslant \theta \leqslant 360^{\circ}

.

[5]