Polynomials

I have no idea how to do this.  So if a mathematician wants to explain it that would be really good.

Of course, if the coefficients did not have to be integers then a could be as close to 0 as you please.

I did construct a graph of degree 7 but it is not actually helpful

Here is a link:

https://www.geogebra.org/classic/tbnjw62b

Using Mathcad to do the heavy lifting I get the following:

I suspect there is a more elegant way, but I can;t think of one!

Thanks Alan,

That works  :)     

Guest's answer is exactly 3 times more than yours.  That suggests more than just a coincidence....