Factorise the polynomial $x^{4} - 5 x^{2} + 4$ completely.

hard 4 marks Class 7 · Mathematics · Algebraic Expressions

Answer

(x-1)(x+1)(x-2)(x+2)

The correct answer is (x-1)(x+1)(x-2)(x+2) — Multiply the factored parts to obtain the complete factorisation: $(x - 1) (x + 1) (x - 2) (x + 2)$ .

Step-by-step solution

Introduce a new variable $y = x^{2}$ to transform the quartic into a quadratic: $y^{2} - 5 y + 4$ .
Factor the quadratic: $y^{2} - 5 y + 4 = (y - 1) (y - 4)$ .
Replace $y$ back with $x^{2}$ : $(x^{2} - 1) (x^{2} - 4)$ .
Factor each difference of squares: $x^{2} - 1 = (x - 1) (x + 1)$ and $x^{2} - 4 = (x - 2) (x + 2)$ .
Multiply the factored parts to obtain the complete factorisation: $(x - 1) (x + 1) (x - 2) (x + 2)$ .

Common mistake: Treating

x^{4} - 5 x^{2} + 4

as a simple quadratic in

x

Concept tested: Factorisation of biquadratic

Published 4 October 2026 · How this solution was prepared and checked

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Factorise the polynomial x4−5x2+4 completely.

Step-by-step solution

Factorise the polynomial $x^{4} - 5 x^{2} + 4$ completely.