Question

Expand: ${(x^2{+y}^2+z^2)}^2$

Answer 1

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

Given: ${(x^2+y^2+z^2)}^2$ we apply the above identity as follows:

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

$=x^4+y^4+z^4+2x^2{y}^2+2y^2{z}^2+2z^2{x}^2$

Hence, ${(x^2+y^2+z^2)}^2=x^4+y^4+z^4+2x^2{y}^2+2y^2{z}^2+2z^2{x}^2$.