Suppose is a cubic polynomial satisfying the following conditions:

MATHEMATICS · Subjective · A Level · Mathematics · Algebra and Functions · Algebra and Functions

Problem

Suppose

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

is a cubic polynomial satisfying the following conditions: (i)

(x+1)

is a factor of

P(x)

. (ii)

(x-2)

is a factor of

P(x)

. (iii) When

P(x)

is divided by

x-3

, the remainder is

5

. (a) Form three equations in

a

b

c

, and

d

based on the above information. (b) Given that

a=1

, solve the equations to determine

b

c

, and

d

. (c) Using the complete polynomial obtained, factorise

P(x)

completely over the real numbers. (d) Write the roots of the equation

P(x)=0

. Provide detailed solutions for each part.