Question

Factorize: 

x⁴ + x² + 25.

Answer 1

We have,

First we rewrite the question,

x⁴ + x² + 25 = (x²)² + 2.x².5 + 52 - 9x²

= {(x²)² + 2.x².5 + 52} – (3x)² 

[By using a² + 2ab + b² = (a + b)²]

= {x² + 5}² – (3x)² 

[ By using a² – b² = (a + b) (a - b)

= (x² + 5 + 3x) ( x² + 5 - 3x)

Thus , 

the factors of

\(=\text{x}^{4}+\text{x}^{2}+25\,are\) \((\text{x}^{2}+5+3\text{x})(\text{x}^{2}+5-3\text{x}).\)