Question

Factorize
$a^3-b^3+1+3ab$

Answer 1

Given : $a^3-b^3+1+3ab$

this can be written as

$=a^3+(-b{)}^3+1^3-3\times a\times (-b)\times 1$

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

$=(a-b+1)[a^2+(-b{)}^2+1-a(-b)-(-b\times 1)-1\times a]$

$=(a-b+1)(a^2+b^2+1+ab+b-a)$

$=(a-b+1)(a^2+b^2+ab-a+b+1)$