Question

Find the product of $(a-b-c)(a^2+b^2+c^2-ab+bc-ca)$ without actual multiplication

Answer 1

$(a+b+c)(a^2+b^2+c^2-ab-bc-ac)$

We know that$(x+y+z)(x^2+y^2+z^2-xy-yz-zx=x^3+y^3+z^3-3xyz$

Comparing both, we get

$x=a,y=b$ and $z=c$

Then $(a+b+c)\left[\right(a{)}^2+(b{)}^2+(c{)}^3-\left(a\right)\left(b\right)-\left(b\right)\left(c\right)-\left(a\right)\left(c\right)]$

$=a^3+b^3+c^3-3\left(a\right)\left(b\right)\left(c\right)$

$=a^3+b^3+c^3-3abc$