Question

If $\frac{a}{b}+\frac{b}{a}=-1$ then find $a^3-b^3$.

Answer 1

$\frac{a}{b}+\frac{b}{a}=-1$

$a^2+b^2=-ab$

$a^2+ab+b^2=0-----\left(1\right)$

$a^3-b^3=(a-b)(a^2+b^2+ab)$

But from $\left(1\right)$

$a^2+b^2+ab=0$

$a^3-b^3=(a-b)\left(0\right)=0$