Without actual division, prove that x⁴ + 2x³ - 2x² + 2x - 3 is exactly divisible by x² + 2x - 3.
Factorisation
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Question

Without actual division, prove that x⁴ + 2x³ - 2x² + 2x - 3 is exactly divisible by x² + 2x - 3.

KnowledgeBoat · Accepted Answer

Let, f(x) = x⁴ + 2x³ - 2x² + 2x - 3 g(x) = x² + 2x - 3 = x² + 3x - x - 3 = x(x + 3) - 1(x + 3) = (x - 1)(x + 3) (x - 1) and (x + 3) are factors of g(x). In order to prove that f(x) is exactly divisible by g(x), it is sufficient to prove that x - 1 and x + 3 are factors of f(x) i.e. it is sufficient to show that f(1) = 0 and f(-3) = 0. Now, f(1) = (1)⁴ + 2(1)³ - 2(1)² + 2(1) - 3 = 1 + 2 - 2 + 2 - 3 = 0 f(-3) = (-3)⁴ + 2(-3)³ - 2(-3)² + 2(-3) - 3 = 81 - 54 - 18 - 6 - 3 = 0 ∴ f(x) is divisible by (x - 1) and (x + 3) Hence, f(x) is exactly divisible by g(x).

Mathematics

Without actual division, prove that x⁴ + 2x³ - 2x² + 2x - 3 is exactly divisible by x² + 2x - 3.

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