Question

Factorize: $2x^2+y^2+8z^2-2\sqrt{2}xy+4\sqrt{2}yz-8zx$.

Answer 1

Factorize the given polynomial

$$\begin{gathered} 2x^2+y^2+8z^2-2\sqrt{2}xy+4\sqrt{2}yz-8zx \\ ={(-\sqrt{2}x)}^2+{\left(y\right)}^2+{\left(2\sqrt{2}z\right)}^2+2\times (-\sqrt{2}x)\times y+2\times y\times \left(2\sqrt{2}z\right)+2\times (-\sqrt{2}x)\times 2\sqrt{2}z) \end{gathered}$$

Use the identity ${(a+b+c)}^2=a^2+b^2+c^2+2ab+2bc+2ca$

$2x^2+y^2+8z^2-2\sqrt{2}xy+4\sqrt{2}yz-8zx={(-\sqrt{2}x+y+2\sqrt{2}z)}^2$

Hence the factorized form of the polynomial $2x^2+y^2+8z^2-2\sqrt{2}xy+4\sqrt{2}yz-8zx$ is $(-\sqrt{2}x+y+2\sqrt{2}z)$$(-\sqrt{2}x+y+2\sqrt{2}z)$.