Question

Factorize $x^4+x^2+1$

Answer 1

Factorize the given expression:

Given expression is $x^4+x^2+1$
Add and subtract $x^2$, we get
$$\begin{gathered} \Rightarrow x^4+2x^2+1-x^2={\left(x^2\right)}^2+2\times x^2\times 1+1^2-x^2 \\ ={\left(x^2+1\right)}^2-{\left(x\right)}^2\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{[}\mathbf{\because }{\mathbf{(}\mathbf{a}\mathbf{+}\mathbf{b}\mathbf{)}}^{\mathbf{2}}\mathbf{=}{\mathbf{a}}^{\mathbf{2}}\mathbf{+}\mathbf{2}\mathbf{a}\mathbf{b}\mathbf{+}{\mathbf{b}}^{\mathbf{2}}\mathbf{]} \\ =\left(x^2+1+x\right)\left(x^2+1-x\right)\mathbf{}\mathbf{[}\mathbf{\because }{\mathbf{a}}^{\mathbf{2}}\mathbf{-}{\mathbf{b}}^{\mathbf{2}}\mathbf{=}\mathbf{(}\mathbf{a}\mathbf{+}\mathbf{b}\mathbf{)}\mathbf{(}\mathbf{a}\mathbf{-}\mathbf{b}\mathbf{)}\mathbf{]} \end{gathered}$$

Hence the required factorized form is $\left(x^2+1+x\right)\left(x^2+1-x\right)$