Question

The expression 4x³ – bx² + x – c leaves remainders 0 and 30 when divided by x + 1 and 2x – 3 respectively. Calculate the values of b and c. Hence, factorise the expression completely. 

Answer 1

given the cubic expression: f(x) = 4x³ − bx² + x − c

Use the Remainder Theorem

Use f(−1) = 0

x + 1 = 0 ⇒ x = −1, we substitute into the expression:

f(−1) = 4(−1)³ −b(−1)² + (−1) − c = 0

−4 − b − 1 − c = 0 ⇒ −b − c = 5 ⇒ b + c = −5

Use `f(3/2) = 30`

Because 2x − 3 = 0 ⇒ `x = 3/2`

`f(3/2) = 4(3/2)^3 - b(3/2)^2 + (3/2) - c = 30`

`(3/2)^3 = 27/8, "so"  4xx 27/8 = 108/8 = 13.5`

`(3/2)^2 = 9/4, bxx 9/4`

`13.5 - b xx 9/4 + 3/2 - c = 30`

`13.5 + 1.5 - c -(9b)/4 = 30 => 15-c-9b/4 = 30`

`-c-(9b)/4 = 15 => c + (9b)/4 (2) -15`

b + c = −5 ⇒ c = −5 − b

`(-5-b) + (9b)/4 = -15 => -5-b+ (9b)/4 = -15`

−20 − 4b + 9b = −60 ⇒ 5b = −40 ⇒ b = −8

b + c = −5 ⇒ −8 + c = −5 ⇒ c = 3

f(x) = 4x³ + 8x² + x − 3

We use factor theorem and trial substitution to find a root.

f(1) = 4 + 8 + 1 − 3 = 10 ≠ 0

f(−1) = −4 + 8 − 1 − 3 = 0

Divide 4x³ + 8x² + x − 3 by (x + 1)

4x² + 4x − 3

4x² + 4x − 3 = (2x − 1) (2x + 3)

4x³ + 8x² + x − 3 = (x + 1) (2x − 1) (2x + 3)