if a/b +b/a =2, then a³ -b³
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Step-by-step explanation:
a
3
−b
3
=0
Step-by-step explanation:
\begin{gathered}Given\\\frac{a}{b}+\frac{b}{a}=-1\end{gathered}
Given
b
a
+
a
b
=−1
Multiply each term by ab , we get
\implies a^{2}+b^{2}=-ab--(1)⟹a
2
+b
2
=−ab−−(1)
\begin{gathered}Now, a^{3}-b^{3}\\=(a-b)(a^{2}+ab+b^{2})\\=(a-b)(a^{2}+b^{2}+ab)\\=(a-b)(-ab+ab)\end{gathered}
Now,a
3
−b
3
=(a−b)(a
2
+ab+b
2
)
=(a−b)(a
2
+b
2
+ab)
=(a−b)(−ab+ab)
/* From (1)*/
=(a-b)\times 0=(a−b)×0
=0=0
Therefore,
a^{3}-b^{3}=0a
3
−b
3
=0
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