Question

If the polynomial f(x) = ax³ + bx – c is divisible by the polynomial g(x) = x² + bx + c, then c =

A. b

B. 2b

C. 2b²

D. – 2b

Answer 1

Given:

f(x) = ax³ + bx – c is divisible by g(x) = x² + bx + c

To find:

The value of c.

Solution:

Since

f(x) = ax³ + bx – c is divisible by g(x) = x² + bx + c

Now by division Method first we have to calculate the value of ab,

As f(x) is divisible by g(x), then remainder must be 0,

i.e. 

(ab² – ac + b)x + abc - c = 0

⇒ (ab² – ac + b)x + c(ab – 1) = 0

⇒ (ab² – ac + b)x = 0 and c(ab – 1) = 0

⇒ ab² – ac + b = 0 (∵ x ≠ 0) and ab – 1 = 0 (∵ c ≠ 0)

⇒ ab – 1 = 0

⇒ ab = 1

Now take;

ab² – ac + b = 0

⇒ (ab)b – ac + b = 0

Using ab = 1 we get;

⇒ b – ac + b = 0

⇒ 2b – ac = 0

⇒ c = 2b/a ... (1)

Multiply and divide RHS of (1) by b

⇒ c = 2b/a = 2b²/ab

As ab = 1

So we get;

c = 2b²