The polynomials f(x) = (ax^3 + 3x^2 − 3) and g(x) = (2x^3 − 5x + a) when divided by (x − 4) leave the same remainder in each case. Find the value of a.

Question

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By remainder theorem, If f(x) is divided by (x - a), then remainder = f(a). Given, f(x) = ax3 + 3x2 - 3 g(x) = 2x3 - 5x + a Divisor : ⇒ x - 4 = 0 ⇒ x = 4 On dividing ax3 + 3x2 - 3 by x - 4, ⇒ f(4) = a(4)3 + 3(4)2 - 3 = 64a + 48 - 3 = 64a + 45. On dividing 2x3 - 5x + a by x - 4, ⇒ g(4) = 2(4)3 - 5(4) + a = 128 - 20 + a = 108 + a. Given, On dividing by (x - 4) polynomials f(x) = ax3 + 3x2 - 3 and g(x) = 2x3 - 5x + a leave same remainder. ⇒ f(4) = g(4) ⇒ 64a + 45 = 108 + a ⇒ 64a - a = 108 - 45 ⇒ 63a = 63 ⇒ a = ⇒ a = 1. Hence, the value of a = 1.

Mathematics

The polynomials f(x) = ax³ + 3x² - 3 and g(x) = 2x³ - 5x + a when divided by (x - 4) leave the same remainder in each case. Find the value of a.

Factorisation

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Mathematics

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Factorisation

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