<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] Another method that requires some intuition and knowledge of cubics:
Observe that, $$\large{\displaystyle{(\color{red}{x + y})^3 = \color{purple}{x}^3 + 3x^2y + 3xy^2 + \color{purple}{y}^3} = \color{purple}{x}^3 + 3\color{green}{xy}\big(\color{red}{x+y}\big) + \color{purple}{y}^3}$$
Otherwise re-arranged, gives:
$$\large{\displaystyle{\color{purple}{x}^3 + \color{purple}{y}^3 =(\color{red}{x + y})^3 - 3\color{green}{xy}\big(\color{red}{x+y}\big)}}$$
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