Question

Find the product using appropriate identity:
$(x+y-z)(x^2+y^2+z^2-xy+yz+zx)$

Answer 1

We know the identity $(a+b+c)(a^2+b^2+c^2-ab-bc-ca)=a^3+b^3+c^3-3abc$

Using the above identity taking $a=x$, $b=y$ and $c=-z$, the product $(x+y-z)(x^2+y^2+z^2-xy+yz+zx)$ can be computed as follows:

$(x+y-z)(x^2+y^2+z^2-xy+yz+zx)=x^3+(y{)}^3+(-z{)}^3-(3x\times y\times -z)=x^3+y^3-z^3+3xyz$

Hence, $(x+y-z)(x^2+y^2+z^2-xy+yz+zx)=x^3+y^3-z^3+3xyz$