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.

Question

KnowledgeBoat · Accepted Answer

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 = Substituting x = in 2x3 + ax2 + bx - 2, will leave remainder 7. 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 = = 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 : 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).

Mathematics

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.

Factorisation

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