If (x + 1) and (x - 2) are factors of x³ + (a + 1)x² - (b - 2)x - 6, find the values of a and b. And then, factorise the given expression completely.
Factorisation
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Question

If (x + 1) and (x - 2) are factors of x³ + (a + 1)x² - (b - 2)x - 6, find the values of a and b. And then, factorise the given expression completely.

KnowledgeBoat · Accepted Answer

Given, (x + 1) and (x - 2) are factors of x³ + (a + 1)x² - (b - 2)x - 6 x + 1 = 0 ⇒ x = -1. Since, x + 1 is factor of x³ + (a + 1)x² - (b - 2)x - 6. Hence, on substituting x = -1 in expression, remainder = 0. ⇒ (-1)³ + (a + 1)(-1)² - (b - 2)(-1) - 6 = 0 -1 + (a + 1)(1) - (-b + 2) - 6 = 0 -1 + a + 1 + b - 2 - 6 = 0 a + b = 8 b = 8 - a ........(i) x - 2 = 0 ⇒ x = 2. Since, x - 2 is factor of x³ + (a + 1)x² - (b - 2)x - 6. Hence, on substituting x = 2 in expression, remainder = 0. ⇒ (2)³ + (a + 1)(2)² - (b - 2)(2) - 6 = 0 ⇒ 8 + (a + 1)(4) - (2b - 4) - 6 = 0 ⇒ 8 + 4a + 4 - 2b + 4 - 6 = 0 ⇒ 10 + 4a - 2b = 0 ⇒ 2b = 4a + 10 ⇒ b = 2a + 5 ........(ii) From (i) and (ii) we get, ⇒ 8 - a = 2a + 5 ⇒ 2a + a = 8 - 5 ⇒ 3a = 3. ⇒ a = 1. Substituting a = 1, in (i) we get, ⇒ b = 8 - 1 = 7. Substituting a = 1, b = 7 in expression we get, Expression = x³ + (1 + 1)x² - (7 - 2)x - 6 = x³ + 2x² - 5x - 6. On dividing, x³ + 2x² - 5x - 6 by (x + 1), we get, quotient = x² + x - 6. Factorising, x² + x - 6 = x² + 3x - 2x - 6 = x(x + 3) - 2(x + 3) = (x - 2)(x + 3). Hence, a = 1, b = 7 and x³ + 2x² - 5x - 6 = (x + 1)(x - 2)(x + 3).

Mathematics

If (x + 1) and (x - 2) are factors of x³ + (a + 1)x² - (b - 2)x - 6, find the values of a and b. And then, factorise the given expression completely.

Factorisation

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Mathematics

If (x + 1) and (x - 2) are factors of x3 + (a + 1)x2 - (b - 2)x - 6, find the values of a and b. And then, factorise the given expression completely.

Factorisation

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