Question

How do you create a polynomial p which has zeros `c=1, c=3`, `c=-3` is a zero of multiplicity 2, the leading term is `-5x^3`?

Answer 1

`-5x^3+5x^2+45x-45`

Explanation:

Note that `x` is a zero of a polynomial if and only if `(x-c)` is a factor of that polynomial.

So in order to have zeros `1`, `3` and `-3`, our polynomial must be a multiple of:

`(x-1)(x-3)(x+3) = (x-1)(x^2-9) = x^3-x^2-9x+9`

In order that its leading term be `-5x^3`, we just need to multiply it by `-5` to get:

`-5(x^3-x^2-9x+9) = -5x^3+5x^2+45x-45`