Polynomial functions and their roots

This problem challenges you to find cubic equations which satisfy different conditions.

Given a set of points (x,y) with distinct x values, find a polynomial that goes through all of them, then prove some results about the existence and uniqueness of these polynomials.

Can you fit polynomials through these points?

Find the relationship between the locations of points of inflection, maxima and minima of functions.

Exploit the symmetry and turn this quartic into a quadratic.

Observe symmetries and engage the power of substitution to solve complicated equations.

How do scores on dice and factors of polynomials relate to each other?

What have Fibonacci numbers to do with solutions of the quadratic equation x^2 - x - 1 = 0 ?

A sequence of polynomials starts 0, 1 and each poly is given by combining the two polys in the sequence just before it. Investigate and prove results about the roots of the polys.

Quadratic graphs are very familiar, but what patterns can you explore with cubics?