Question

Find a polynomial equation that has zeros at x = −2, x = 0, x = 3 and x = 5

Answer 1

If a polynomial "contains", in a multiplicative sense, a factor $(x-x_0)$, then the polynomial has a zero at $x=x_0$.

So, you polynomial must contain at least the following:

$(x-(-2)),\quad (x-0),\quad (x-3),\quad (x-5)$

If you multiply them all, you get

$x(x+2)(x-3)(x-5)=x^4 - 6 x^3 - x^2 + 30 x$

Now, if you want the polynomial to be zero only and exactly at the four points you've given, you can choose every polynomial that is a multiple (numerically speaking) of this one. For example, you can multiply it by 2, 3, or -14.

If you want the polynomial to be zero at least at the four points you've given, you can multiply the given polynomial by every other function.