If (x3 + ax2 + bx + 6) has (x - 2) as a factor and leaves a remainder 3 when divided by (x - 3), find the values of a and b.

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KnowledgeBoat · Accepted Answer

Let f(x) = x3 + ax2 + bx + 6 By factor theorem, If, (x - 2) is a factor of f(x), then f(2) = 0. ⇒ (2)3 + a(2)2 + b(2) + 6 = 0 ⇒ 8 + 4a + 2b + 6 = 0 ⇒ 4a + 2b + 14 = 0 ⇒ 2(2a + b + 7) = 0 ⇒ 2a + b + 7 = 0 ⇒ 2a + b = -7 ....(1) Given, On dividing f(x) by (x − 3), the remainder is 3. By remainder theorem, ∴ f(3) = 3 ⇒ (3)3 + a(3)2 + b(3) + 6 = 3 ⇒ 27 + 9a + 3b + 6 = 3 ⇒ 9a + 3b + 33 = 3 ⇒ 9a + 3b = 3 - 33 ⇒ 9a + 3b = -30 ⇒ 3(3a + b) = -30 ⇒ 3a + b = ⇒ 3a + b = -10 ....(2) Subtracting equation (1) from equation (2), ⇒ 3a + b - (2a + b) = -10 -(-7) ⇒ 3a + b - 2a - b = -10 + 7 ⇒ a = -3. Substituting value of a in equation (1), we get : ⇒ 2(-3) + b = -7 ⇒ -6 + b = -7 ⇒ b = -7 + 6 ⇒ b = -1. Hence, the value of a = -3 and b = -1.

Mathematics

If (x³ + ax² + bx + 6) has (x - 2) as a factor and leaves a remainder 3 when divided by (x - 3), find the values of a and b.

Factorisation

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Mathematics

If (x3 + ax2 + bx + 6) has (x - 2) as a factor and leaves a remainder 3 when divided by (x - 3), find the values of a and b.

Factorisation

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