Question

Factorise $x^4-625$.

Answer 1

Factorize the given polynomial

Given: $x^4-625$

$\Rightarrow x^4-625={\left(x^2\right)}^2-{\left(25\right)}^2$

Step 1: use the identity $a^2-b^2=(a-b)(a+b)$.

$$\begin{gathered} \Rightarrow x^4-625=(x^2-25)(x^2+25) \\ \Rightarrow x^4-625=\left[{\left(x\right)}^2-{\left(5\right)}^2\right](x^2+25) \end{gathered}$$

Step 2: Factorize further

$\Rightarrow x^4-625=(x-5)(x+5)(x^2+25)$

Hence the factorized form of $x^4-625$ is $(x-5)(x+5)(x^2+25)$.