Question

Factorise $\left(x^2-2xy+y^2\right)-z^2$

Answer 1

Factorizing the expression:;

$\left(x^2-2xy+y^2\right)-z^2$

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

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

Hence, the factors of $\left(x^2-2xy+y^2\right)-z^2$ are $\left(x-y-z\right)$ and $\left(x-y+z\right)$.