The polynomial 3x^3 + 8x^2 - 15x + k has (x - 1) as a factor. Find the value of k. Hence factorize the resulting polynomial completely.

Question

KnowledgeBoat · Accepted Answer

⇒ x - 1 = 0 ⇒ x = 1. Given, (x - 1) is a factor of 3x3 + 8x2 - 15x + k. Thus, on substituting x = 1 in 3x3 + 8x2 - 15x + k, the remainder will be zero. ⇒ 3.(1)3 + 8.(1)2 - 15(1) + k = 0 ⇒ 3.1 + 8.1 - 15 + k = 0 ⇒ 3 + 8 - 15 + k = 0 ⇒ 11 - 15 + k = 0 ⇒ k - 4 = 0 ⇒ k = 4. Polynomial = 3x3 + 8x2 - 15x + 4 On dividing (3x3 + 8x2 - 15x + 4) by (x - 1), we get : ⇒ 3x3 + 8x2 - 15x + 4 = (x - 1)(3x2 + 11x - 4) = (x - 1)[3x2 + 12x - x - 4] = (x - 1)[3x(x + 4) - 1(x + 4)] = (x - 1)(3x - 1)(x + 4). Hence, 3x3 + 8x2 - 15x + 4 = (x - 1)(3x - 1)(x + 4).

Mathematics

The polynomial 3x³ + 8x² - 15x + k has (x - 1) as a factor. Find the value of k. Hence factorize the resulting polynomial completely.

Factorisation

ICSE 2024

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