Constructing a polynomial with certain zeroes. I want to construct a polynomial $f(x)$ that has zeroes at $-9,\,-5,\,0,\, 5,\, 9$.

Can somebody provide a method (or perhaps some hints) for solving this?
And nowhere else, I guess, otherwise the zero polynomial will do. And I assume your are working over the rationals, say.

Just use the fact that $f(x)$ has a zero at $a$ if and only if $x - a$ divides $f(x)$.
Try this:

$$f(x):=(x-5)(x+5)(x-9)(x+9)x$$