The expression 2x³ + ax² + bx - 2 leaves remainder 7 and 0 when divided by 2x - 3 and x + 2 respectively. Calculate the values of a and b.

2x - 3 = 0 ⇒ x = $\dfrac{3}{2}$

Given, when 2x³ + ax² + bx - 2 is divided by 2x - 3, the remainder is 7.

∴ On substituting x = $\dfrac{3}{2}$ in 2x³ + ax² + bx - 2 , remainder = 7.

$\Rightarrow 2\Big(\dfrac{3}{2}\Big)^3 + a\Big(\dfrac{3}{2}\Big)^2 + b\Big(\dfrac{3}{2}\Big) - 2 = 7\\[1em] \Rightarrow 2\Big(\dfrac{27}{8}\Big) + a\Big(\dfrac{9}{4}\Big) + b\Big(\dfrac{3}{2}\Big) - 2 = 7 \\[1em] \Rightarrow \dfrac{27}{4} + \dfrac{9a}{4} + \dfrac{3b}{2} - 2 = 7\\[1em] \Rightarrow \dfrac{27 + 9a + 6b - 8}{4} = 7 \\[1em] \Rightarrow 9a + 6b + 19 = 28 \\[1em] \Rightarrow 9a + 6b = 28 - 19 \\[1em] \Rightarrow 3(3a + 2b) = 9 \\[1em] \Rightarrow 3a + 2b = 3 \\[1em] \Rightarrow 2b = 3 - 3a ……..(i)$

x + 2 = 0 ⇒ x = -2

Given, when 2x³ + ax² + bx - 2 is divided by x + 2, the remainder is 0.

∴ On substituting x = -2 in 2x³ + ax² + bx - 2 , remainder = 0.

⇒ 2(-2)³ + a(-2)² + b(-2) - 2 = 0

⇒ 2(-8) + 4a - 2b - 2 = 0

⇒ -16 + 4a - 2b - 2 = 0

⇒ 4a - 2b - 18 = 0

Substituting value of 2b from (i) in above equation,

⇒ 4a - (3 - 3a) - 18 = 0

⇒ 4a + 3a - 18 - 3 = 0

⇒ 7a = 21

⇒ a = 3.

Substituting value of a in (i) we get,

⇒ 2b = 3 - 3(3)

⇒ 2b = 3 - 9

⇒ 2b = -6

⇒ b = -3.

Hence, a = 3 and b = -3.