If $x+\frac1x=-5$, what is $x^3+\frac1{x^3}$?
To solve this problem, I will use the identity $(x+\frac1x)^3=x^3+\frac1{x^3}+3(x+\frac1x)$, which I can prove by expanding the left-hand side and simplifying. Applying this identity to the given equation, I get $(-5)^3=x^3+\frac1{x^3}+3(-5)$. Simplifying this equation, I get $-125=x^3+\frac1{x^3}-15$. Adding 15 to both sides, I get $x^3+\frac1{x^3}=-110$. 