Question

Prove the following identities:
$(y+z{)}^2(z+x{)}^2(x+y{)}^2=\sum x^4(y+z{)}^2+2(\sum yz{)}^3-2x^2{y}^2{z}^2$.

Answer 1

$\Rightarrow$ Since $(y+z)(z+x)(x+y)$

$=x^2(y+x)+y^2(z+x)+z^2(x+y)+2xyz$

We have, ${(y+z)}^2{(z+x)}^2{(x+y)}^2$

$=\sum x^4{(y+z)}^2+2x^2{y}^2(z+x)(y+x)+2x^2{z}^2(x+y)(z+y)$

$+2y^2{z}^2(z+x)(y+x)+4x^{3}yz(y+z)+4x{y}^{3}z(z+x)+4xy{z}^3(x+y)+4x^2{y}^2{z}^2$

$=\sum x^4{(y+z)}^2+2\sum x^3{y}^3+6\sum x^3{y}^{2}z+10x^2{y}^2{z}^2$

$=\sum x^4{(y+z)}^2+2(\sum x^3{y}^3+3\sum x^3{y}^{2}z+6x^2{y}^2{z}^2)-2x^2{y}^2{z}^2$

$=\sum x^4{(y+z)}^2+2{(\sum xy)}^3-2x^2{y}^2{z}^2$