If (x − 2) is a factor of the expression 2x^3 + ax^2 + bx − 14 and when the expression is divided by (x − 3), it leaves a remainder 52, find the values of a and b.

Question

KnowledgeBoat · Accepted Answer

Let f(x) = 2x3 + ax2 + bx - 14 Since, (x − 2) is a factor of f(x) then f(2) = 0. ⇒ 2(2)3 + a(2)2 + b(2) - 14 = 0 ⇒ 2(8) + 4a + 2b - 14 = 0 ⇒ 16 + 4a + 2b - 14 = 0 ⇒ 4a + 2b + 2 = 0 ⇒ 4a + 2b = -2 ⇒ 2(2a + b) = -2 ⇒ 2a + b = ⇒ 2a + b = -1 ....(1) On dividing f(x) by (x − 3), the remainder is 52, By remainder theorem, ⇒ f(3) = 52 ⇒ 2(3)3 + a(3)2 + b(3) - 14 = 52 ⇒ 2(27) + 9a + 3b - 14 = 52 ⇒ 54 + 9a + 3b - 14 = 52 ⇒ 9a + 3b + 40 = 52 ⇒ 9a + 3b = 52 - 40 ⇒ 9a + 3b = 12 ⇒ 3(3a + b) = 12 ⇒ 3a + b = ⇒ 3a + b = 4 ....(2) Subtracting equation (1) from (2), we get : ⇒ 3a + b - (2a + b) = 4 - (-1) ⇒ a = 4 + 1 ⇒ a = 5. Substituting a = 5 in equation (1), we get : ⇒ 2(5) + b = -1 ⇒ 10 + b = -1 ⇒ b = -1 - 10 ⇒ b = -11. Hence, the value of a = 5 and b = -11.

Mathematics

If (x - 2) is a factor of the expression 2x³ + ax² + bx - 14 and when the expression is divided by (x - 3), it leaves a remainder 52, find the values of a and b.

Factorisation

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If (x - 2) is a factor of the expression 2x3 + ax2 + bx - 14 and when the expression is divided by (x - 3), it leaves a remainder 52, find the values of a and b.

Factorisation

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