Question:
If $\frac{a}{b}+\frac{b}{a}=-1$ then $\left(a^{3}-b^{3}\right)=?$
(a) −3
(b) −2
(c) −1
(d) 0
Solution:
$\frac{a}{b}+\frac{b}{a}=-1$
$\Rightarrow \frac{a^{2}+b^{2}}{a b}=-1$
$\Rightarrow a^{2}+b^{2}=-a b$
$\Rightarrow a^{2}+b^{2}+a b=0$
Thus, we have:
$\left(a^{3}-b^{3}\right)=(a-b)\left(a^{2}+b^{2}+a b\right)$
$=(a-b) \times 0$
$=0$
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