Question

Factorize the following expression: $x^4-{(y+z)}^4$.

Answer 1

Factorize${\mathit{x}}^{\mathbf{4}}\mathbf{-}\mathbf{}{\mathbf{(}\mathbf{y}\mathbf{}\mathbf{+}\mathbf{}\mathbf{z}\mathbf{)}}^{\mathbf{4}}$.

The given expression can be written as

$x^4-{(y+z)}^4={\left(x^2\right)}^2-{\left({\left(y+z\right)}^2\right)}^2$

$=\left(x^2+{\left(y+z\right)}^2\right)\left(x^2-{\left(y+z\right)}^2\right)$ $\left[\therefore a^2-b^2=(a+b)(a-b)\right]$

$=\left(x^2+y^2+2yz+z^2\right)\left(x^2-y^2-2yz-z^2\right)$ $\left[\therefore {\left(a+b\right)}^2=a^2+2ab+b^2\right]$

$$\begin{gathered} =\left(x^2+y^2+z^2+2yz\right)\left(x^2-y^2-z^2-2yz\right) \\ =\left(x^2+y^2+z^2+2yz\right)\left(x+y+z\right)\left(x-y-z\right) \end{gathered}$$

Hence, the factorized form is$\left(x^2+y^2+z^2+2yz\right)\left(x+y+z\right)\left(x-y-z\right)$.