Question
If $\frac{\text{a}}{\text{b}}+\frac{\text{b}}{\text{a}}=-1,$ then $a^3 - b^3 =$
✓
Answer
$\frac{\text{a}}{\text{b}}+\frac{\text{b}}{\text{a}}=-1$
$\Rightarrow\frac{\text{a}^2+\text{b}^2}{\text{ab}}=-1$
$\Rightarrow\text{a}^2+\text{b}^2+\text{ab}=0$
Now using identity
$a^3 - b^3$
$= (a - b)(a^2 + b^2 + ab)$
$=(a - b)(0) (\because a^2 + b^2 + ab = 0)$
$= 0$
$\Rightarrow\frac{\text{a}^2+\text{b}^2}{\text{ab}}=-1$
$\Rightarrow\text{a}^2+\text{b}^2+\text{ab}=0$
Now using identity
$a^3 - b^3$
$= (a - b)(a^2 + b^2 + ab)$
$=(a - b)(0) (\because a^2 + b^2 + ab = 0)$
$= 0$
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