Systems of Equations · Challenge problem

For every pair of numbers $x$ and $y$ with $x + y = a$ and $xy = b$, express each of the following in terms of $a$ and $b$ alone.

$(a)\ x^2 + y^2$ $(b)\ \dfrac{1}{x} + \dfrac{1}{y}$ $(c)\ x^3 + y^3$