Binomial Expansion based on equation for evaluation

Question

I have this question that has really stumped me, it is supposed to be done via Binomial kind of expansion.

If $x+\frac1x=10$ find the value of $x^3+\frac1{x^3}$.

So I hope some one has an approach to this question.

Answer 1

Note that $$ \left(x+\frac{1}{x}\right)^3=x^3+\frac{1}{x^3}+3x^2\left( \frac{1}{x} \right)+3x\left( \frac{1}{x^2} \right)=x^3+\frac{1}{x^3}+3\left(x+\frac{1}{x}\right) $$ You are given enough information to solve for the required quantity.

Answer 2

Hint:

Expand $\;\biggl(x+\dfrac1x\biggr)^3$ by the standard formula.

Answer 3

Not sure how to do this with a Binomial expansion but:

$x+\frac1x=10 \to x^2+1=10x \to x^2-10x+1=0$

Solve for $x$ using quadratic formula and plug it into $x^3+\frac1{x^3}$.