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Mathematics

Factorise the following:

x4 - 14x2 + 1

Factorisation

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Answer

Above terms can be written as,

x4 + 2x2 - 16x2 + 1

= x4 + 2x2 + 1 - 16x2

= [(x2)2 + 2x2 + 1] - (4x)2

We know that,

(a + b)2 = a2 + 2ab + b2

and

a2 - b2 = (a - b)(a + b)

∴ [(x2)2 + 2x2 + 1] - (4x)2 = (x2 + 1)2 - (4x)2

= (x2 + 1 - 4x)(x2 + 1 + 4x).

Hence, x4 - 14x2 + 1 = (x2 + 4x + 1)(x2 - 4x + 1).

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