The expression 2x³ + ax² + bx - 2 leaves remainders 0 and 7 when divided by (x + 2) and (2x - 3), respectively. Calculate the values of a and b. Factorise the expression completely.

Given,

The expression 2x³ + ax² + bx - 2 leaves remainders 0 and 7 when divided by (x + 1) and (2x - 3) respectively.

⇒ x + 2 = 0

⇒ x = -2.

Substituting x = -2 in 2x³ + ax² + bx - 2, will leave remainder 0.

⇒ 2(-2)³ + a(-2)² + b(-2) - 2 = 0

⇒ 2(-8) + a(4) - 2b - 2 = 0

⇒ -16 + 4a - 2b - 2 = 0

⇒ 4a - 2b - 18 = 0

⇒ 2b = 4a - 18

⇒ 2b = 2(2a - 9)

⇒ b = 2a - 9 …………..(1)

Taking,

2x - 3 = 0

⇒ 2x = 3

⇒ x = $\dfrac{3}{2}$

Substituting x = $\dfrac{3}{2}$ in 2x³ + ax² + bx - 2, will leave remainder 7.

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

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

⇒ 9a + 6(2a - 9) = 9

⇒ 9a + 12a - 54 = 9

⇒ 21a = 9 + 54

⇒ 21a = 63

⇒ a = $\dfrac{63}{21}$ = 3.

Substituting value of a in equation (1), we get :

⇒ b = 2a - 9 = 2(3) - 9 = 6 - 9 = -3.

Expression = 2x³ + 3x² - 3x - 2.

Dividing 2x³ + 3x² - 3x - 2 by (x + 2), we get :

$\begin{array}{l} \phantom{x + 2)}{\quad 2x^2 - x - 1} \ x + 2\overline{\smash{\big)}\quad 2x^3 + 3x^2 - 3x - 2} \ \phantom{x + 2)}\phantom{)}\underline{\underset{-}{+}2x^3 \underset{-}{+}4x^2} \ \phantom{{x + 2}x^3 + x^2}-x^2 - 3x \ \phantom{{x + 2)}2x^3-2}\underline{\underset{+}{-}x^2 \underset{+}{-} 2x} \ \phantom{{x + 2)}{23x^3-2x^{2}3}}-x - 2 \ \phantom{{x + 2)}{23x^3-2x^{2}(3}}\underline{\underset{+}{-}x \underset{+}{-} 2} \ \phantom{{x + 2)}{23x^3-2x^{2}(31)}3}\times \end{array}$

2x³ + 3x² - 3x - 2 = (x + 2)(2x² - x - 1)

= (x + 2)[2x² - 2x + x - 1]

= (x + 2)[2x(x - 1) + 1(x - 1)]

= (x + 2)(2x + 1)(x - 1).

Hence, a = 3, b = -3 and 2x³ + 3x² - 3x - 2 = (x + 2)(2x + 1)(x - 1).