Given $x+y$ and $x\cdot y$, what is $x^3+ y^3$ ? I have been looking at an assortment of high school number sense tests and I noticed a reoccurring problem that states what x+y is and what $x\cdot y$ is then asks for $x^3+ y^3$. I want to know how to work these problems. I have a couple of examples.
$x+y=5$ and $x\cdot y=1$, then $x^3+y^3=?$ [key says 110]
$x+y=-1$ and $x\cdot y=2$, then $x^3+y^3=?$ [key says 5]
$x-y=-1$ and $x\cdot y=2$, then $x^3 -y^3=?$ [key says -7]
$x+y=\frac{1}{3}$ and $x\cdot y=\frac{1}{9}$, then $x^3+y^3=?$ $\left[-\dfrac{2}{27}\right]$

The straight way is as given by @AndréNicolas, cleverly exploiting remarkable identities.
Alternatively, as we know the sum, let $s$, and the product, let $p$, by Vieta's formulas, we know that the two numbers are the roots of
$$t^2-st+p=0.$$
Then
$$t^3=t\cdot t^2=st^2-pt=s(st-p)-pt=(s^2-p)t-sp$$ and the sum of the cubes of the roots must be
$$(s^2-p)s-2sp=s^2-3sp.$$
