Question

Factor the expression completely.

$x^4-2x^2+1$

Short answer

Factorized form of$x^4-2x^2+1$is ${\left(x-1\right)}^2{\left(x+1\right)}^2$.

Step-by-step solution

Step 1. Recognizing the identity of a whole square.

It is known that,

$\left(a-b\right)=a^2-2ab+b^2$

Step 2. Re write the expression after comparing it with the identity.

$\begin{array}{c}{x}^4-2x^2+1={\left(x^2\right)}^2-2\left(x^2\right)\left(1\right)+{\left(1\right)}^2\\ ={\left(x^2-1\right)}^2\end{array}$

Step 3. Factorize the expression using the identity mentioned below.

$\left(a^2-b^2\right)=\left(a-b\right)\left(a+b\right)$

Hence the simplified expression becomes,

$\begin{array}{c}{\left(x^2-1^2\right)}^2={\left[\left(x-1\right)\left(x+1\right)\right]}^2\\ ={\left(x-1\right)}^2{\left(x+1\right)}^2\end{array}$