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 $x^3+y^3=?$ $\left[-\dfrac{2}{27}\right]$

Hrm. How can we make $x^3 + y^3$ out of that?
What tools do we have available to us? We have $x+y$. We have $xy$. And we have arithmetic.
I don't know yet how to make $x^3 + y^3$ exactly. Let's focus on one part: let's make $x^3$, and then see what we can do with what's left over.
It's pretty easy to make $x^3$ by multiplying things together. We could take $(xy) \cdot (xy) \cdot (xy)$. We could take $(x+y) \cdot(x+y) \cdot (xy)$. There are others. Let's experiment and actually try all of them. Multiplying them out, we can make


*

*$x^3 y^3$

*$x^3 y^2 + x^2 y^3$

*$x^3 y + 2 x^2 y^2 + x y^3$

*$x^3 + 3 x^2 y + 3 x y^2 + y^3$


That last one looks intriguing: it has $x^3$ all by itself without other terms multiplied in. It even has $y^3$ in it too.
So let's go with that, and try to subtract off the rest. Now, the question is

How do we make $3x^2 y + 3 x y^2$

If we can solve that, then we can combine that answer with the above to make $x^3 + y^3$.
