Factorize x⁴ + x² + 25
Question:
Factorize: \[ x^4+x^2+25 \]
Solution
We write
\[ x^4+x^2+25 = x^4+10x^2+25-9x^2 \]
\[ = (x^2+5)^2-(3x)^2 \]
Using the identity
\[ a^2-b^2=(a-b)(a+b) \]
Therefore,
\[ x^4+x^2+25 = (x^2+5-3x)(x^2+5+3x) \]
\[ = (x^2-3x+5)(x^2+3x+5) \]
Therefore,
\[ \boxed{(x^2-3x+5)(x^2+3x+5)} \]