Solve (x+y)(x-y)+(y+z)(y-z)+(z+x)(z-x)

$$( x + y ) ( x - y ) + ( y + z ) ( y - z ) + ( z + x ) ( z - x )$$

Evaluate

$$0$$

Solution Steps

Consider \left(x+y\right)\left(x-y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Consider \left(y+z\right)\left(y-z\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Combine -y^{2} and y^{2} to get 0.

$$x^{2}-z^{2}+\left(z+x\right)\left(z-x\right)$$

Consider \left(z+x\right)\left(z-x\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$x^{2}-z^{2}+z^{2}-x^{2}$$

Combine -z^{2} and z^{2} to get 0.

$$x^{2}-x^{2}$$

Combine x^{2} and -x^{2} to get 0.

$$0$$

Factor

$$0$$