Question

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

Answer 1

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

The given expression can be written as

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

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

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

Hence the factorized form is$\mathbf{z}\mathbf{(}\mathbf{2}\mathbf{x}\mathbf{-}\mathbf{z}\mathbf{)}\mathbf{(}\mathbf{2}{\mathbf{x}}^{\mathbf{2}}\mathbf{-}\mathbf{2}\mathbf{x}\mathbf{z}\mathbf{+}{\mathbf{z}}^{\mathbf{2}}\mathbf{)}$.