Maths Polynomials question?

Question

Here is the question:

Let P(x) = 3x^3-4x-k. If P(x) is divisible by x-k, the remainder when P(x) is divided by x+k is:

A. 2k

B. k

C. 0

D. -k

or

E. -2k

The answer is apparently E. but I have no idea how to get the answer. Please show all workings. Thank you so much! :D

Answer 1

Let P(x) = 3x^3 - 4x - k

if P(x) is divisible by x - k Then

3x^3 - 4x - k is divisible by x - k

x - k = 0

Add k to each side

x = k

Substitute x = k  in the P(x) = 3x^3 - 4x - k

3k^3 - 4k - k

3K^3 - 5k

P(x) is divisible by x + k then

x + k = 0

Subtract k from each side

x = - k

Substitute x = -k in the P(x) = 3x^3 - 4x - k

3(-k)^3 -4(-k) - k

-3k^3 + 4k - k

Simplify

3k - 3k^3

-(3k^3 - 3k)

-3k^3 + 5k - 2k

-(3k^3 - 5k) - 2k

If P(x) is divisible by x-k, the remainder when P(x) is divided by x+k is: -2k.

E is the right choice.