I'm trying to find a way to create an equation of a polynomial based on its turning points. I know the basic equation for quadratics, and found a different one for cubics
$f\left(x\right)=\left(-\frac{6\left(b-d\right)}{\left(a-c\right)^{3}}\left(\frac{x^{3}}{3}-\frac{\left(a+c\right)x^{2}}{2}+acx\right)+\frac{\left(b+d\right)}{2}-\frac{\left(b-d\right)\left(a+c\right)\left(a^{2}+c^{2}-4ac\right)}{2\left(a-c\right)^{3}}\right)$ marty cohen (https://math.stackexchange.com/users/13079/marty-cohen), Find equation of cubic from turning points, URL (version: 2020-02-04): https://math.stackexchange.com/q/3534502
Other than that, is there a simple-ish way to create a polynomial equation given multiple turning points? Say I have turning points $(a,b)$, $(c,d)$, and $(e,f)$. Is there any way to turn these turning points into a polynomial? What if I also have more points, how could I turn those points into the polynomial as well? Is there a simple equation to plug in these values, or are there steps I need to take?
I can't find anything else on the internet about this, so I'm hoping for some help here. Thanks to anyone who can help!