Question

How do you create a polynomial `p` which has zeros `x=+-3, x=6`, leading term is `7x^5`, and the point `(-3,0)` is a local minimum on the graph of `y=p(x)`?

Answer 1

`f(x) = 7x^5+21x^4-315x^3-945x^2+2268x+6804`

Explanation:

Each zero `x=a` corresponds to a linear factor `(x-a)`

We need the zero at `(-3, 0)` to be of even multiplicity in order that it is also a local minimum or maximum.

Also, since the leading coefficient is positive and the point `(-3, 0)` is a local minimum, there must be another real zero less than `-3`.

So let's write:

`f(x) = 7(x+6)(x+3)(x+3)(x-3)(x-6)`

`color(white)(f(x)) = 7(x+3)(x^2-36)(x^2-9)`

`color(white)(f(x)) =7x^5+21x^4-315x^3-945x^2+2268x+6804`

graph{7x^5+21x^4-315x^3-945x^2+2268x+6804 [-10, 10, -12000, 10000]}