Given that x + 2 and x - 3 are factors of x³ + ax + b. Calculate the values of a and b. Also, find the remaining factor.
Factorisation
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Question

Given that x + 2 and x - 3 are factors of x³ + ax + b. Calculate the values of a and b. Also, find the remaining factor.

KnowledgeBoat · Accepted Answer

By factor theorem, If (x - a) is a factor of polynomial f(x), then remainder f(a) = 0. ⇒ x + 2 = 0 ⇒ x = -2. Since, x + 2 is a factor of x³ + ax + b ∴ Remainder = 0. ⇒ (-2)³ + (-2)a + b = 0 ⇒ -8 - 2a + b = 0 ⇒ b = 2a + 8 ................(1) x - 3 = 0 ⇒ x = 3. Since, x - 3 is a factor of x³ + ax + b ∴ Remainder = 0. ⇒ (3)³ + (3)a + b = 0 ⇒ 27 + 3a + b = 0 ⇒ b = -27 - 3a ................(2) From (1) and (2), we get: ⇒ 2a + 8 = -27 - 3a ⇒ 2a + 3a = -27 - 8 ⇒ 5a = -35 ⇒ a = ⇒ a = -7. Substituting value of a in equation (1), we get : ⇒ b = 2a + 8 = 2 × -7 + 8 = -14 + 8 = -6. f(x) = x³ + ax + b = x³ - 7x - 6. Substituting x = -1 in f(x), we get : ⇒ (-1)³ - 7(-1) - 6 ⇒ -1 + 7 - 6 ⇒ -1 + 1 ⇒ 0. ∴ (x + 1) is a factor of x³ - 7x - 6. Hence, a = -7, b = -6 and (x + 1) is the remaining factor.

Mathematics

Given that x + 2 and x - 3 are factors of x³ + ax + b. Calculate the values of a and b. Also, find the remaining factor.

Factorisation

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Mathematics

Given that x + 2 and x - 3 are factors of x3 + ax + b. Calculate the values of a and b. Also, find the remaining factor.

Factorisation

Related Questions

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