<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] You can use the quadratic formula, or use a trick. Recall that $x^3+y^3=(x+y)(x^2-xy+y^2) $.
Now consider that $x^2-xy+y^2=(x+y)^2-3xy$
So you have:
$$x^3+y^3=(x+y)(x^2-xy+y^2)=(x+y)((x+y)^2-3xy)  $$
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