Question

Factorise: $x^4+x^2+1$

Answer 1

Factorize the polynomial

Given: $x^4+x^2+1$

adding and subtracting $x^2$to the polynomial

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

using the identity ${(a+b)}^2=a^2+b^2+2ab$

$={(x^2+1)}^2-x^2$

Using the identity $a^2-b^2=(a-b)(a+b)$

$=(x^2+1-x)(x^2+1+x)$

Hence, the factorized form of the polynomial $x^4+x^2+1$ is $(x^2+1-x)$$(x^2+1+x)$.