The expression 4x³ - bx² + x - c leaves remainder 0 and 30 when divided by (x + 1) and (2x - 3) respectively. Calculate the values of b and c.

By remainder theorem,

When a polynomial p(x) is divided by a linear polynomial (x - a), then the remainder is equal to p(a).

⇒ x + 1 = 0

⇒ x = -1

Given,

4x³ - bx² + x - c leaves remainder 0 on dividing it by (x + 1).

∴ 4(-1)³ - b(-1)² + (-1) - c = 0

⇒ 4(-1) - b(1) - 1 - c = 0

⇒ -4 - b - 1 - c = 0

⇒ b + c = -5

⇒ b = -5 - c ………(1)

Given,

4x³ - bx² + x - c leaves remainder 30 on dividing it by (2x - 3).

2x - 3 = 0

⇒ 2x = 3

⇒ x = $\dfrac{3}{2}$

Substituting x = $\dfrac{3}{2}$ in 4x³ - bx² + x - c should give result as 30,

$\Rightarrow 4 \times \Big(\dfrac{3}{2}\Big)^3 - b \times \Big(\dfrac{3}{2}\Big)^2 + \Big(\dfrac{3}{2}\Big) - c = 30 \\[1em] \Rightarrow 4 \times \Big(\dfrac{27}{8}\Big) - b \times \Big(\dfrac{9}{4}\Big) + \Big(\dfrac{3}{2}\Big) - c = 30 \\[1em] \Rightarrow \dfrac{27}{2} - \dfrac{9b}{4} + \dfrac{3}{2} - c = 30 \\[1em] \Rightarrow \dfrac{54 - 9b + 6 - 4c}{4} = 30 \\[1em] \Rightarrow 60 - 9b - 4c = 120$

Substituting value of b from (1) in above equation :

⇒ 60 - 9(-5 - c) - 4c = 120

⇒ 60 + 45 + 9c - 4c = 120

⇒ 5c + 105 = 120

⇒ 5c = 120 - 105

⇒ 5c = 15

⇒ c = $\dfrac{15}{5}$ = 3.

Substituting value of c in (1), we get :

⇒ b = -5 - c = -5 - 3 = -8.

Hence, b = -8 and c = 3.