<s>[INST] Finding $x^3+y^3$, given $x + y = 5$ and $xy = 1$ 
Given
$$x + y = 5 \qquad xy = 1$$
Find $x^3 + y^3$.

To solve this, I tried this:
$y = \frac{1} {x}$
$x + \frac{1}{x} = 5$
$x^2 - 5x + 1 = 0$
What formula needs to be used to find the value of $x$?
 [/INST] Use the binomial formula. We have
$$(x+y)^3=x^3 + 3x^2y + 3xy^2 + y^3,$$
which reduces to
$$125=x^3 + 3\cdot 5 + y^3$$
and thus
$$x^3 + y^3 = 110.$$
There is no need to solve for $x$.
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