+0

+1
288
2
+606

If $x+\frac1x=-5$, what is $x^3+\frac1{x^3}$?

Apr 9, 2018

#1
+97599
+2

$$x+\frac1x=-5, \;\;\;what \;\;\;is\;\;\;x^3+\dfrac1{x^3}\\~\\ \left(x+\frac{1}{x}\right)^3=x^3+3x^2*\frac{1}{x}+3x*\frac{1}{x^2}+\frac{1}{x^3}\\ \left(x+\frac{1}{x}\right)^3=x^3+3x+\frac{3}{x}+\frac{1}{x^3}\\ \left(x+\frac{1}{x}\right)^3=x^3+\frac{1}{x^3}+3(x+\frac{1}{x})\\ sub\\ (-5)^3=x^3+\frac{1}{x^3}+3(-5)\\ -125=x^3+\frac{1}{x^3}-15\\ -110=x^3+\frac{1}{x^3}\\ x^3+\frac{1}{x^3}=110$$

.
Apr 9, 2018
#2
+606
0

You were almost correct! It was actually -110. Thank you for the help:

To get $x^3$ and $\frac1{x^3}$, we cube $x+\frac1x$: $$-125=(-5)^3=\left(x+\frac1x\right)^3=x^3+3x+\frac3x+\frac1{x^3}$$ by the Binomial Theorem. Conveniently, we can evaluate $3x + \frac{3}{x}$ as $3\left(x+\frac1x\right)=3(-5)=-15$, so $$x^3+\frac1{x^3}=(-125)-(-15)=\boxed{-110}.$$

.
Apr 9, 2018