Question

Factorize the following: ${(x^2+y^2-z^2)}^2-4x^2{y}^2$.

Answer 1

Factorize${({\mathrm{x}}^2+{\mathrm{y}}^2-{\mathrm{z}}^2)}^2-4x^2{y}^2$.

The given expression can be written as

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

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

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

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

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

$=\left(x+y-z\right)\left(x+y+z\right)\left(x-y-z\right)\left(x-y+z\right)$

Hence, the factored expression is $(\mathrm{x}+\mathrm{y}-\mathrm{z})(\mathrm{x}+\mathrm{y}+\mathrm{z})(\mathrm{x}-\mathrm{y}-\mathrm{z})(\mathrm{x}-\mathrm{y}+\mathrm{z})$.