Polynomial functions and their roots

problem
Curve fitter

Age

14 to 18

Challenge level

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

Age

16 to 18

Challenge level

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.
problem
Polynomial interpolation

Age

16 to 18

Challenge level

Can you fit polynomials through these points?
problem
Patterns of inflection

Age

16 to 18

Challenge level

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

Age

16 to 18

Challenge level

Exploit the symmetry and turn this quartic into a quadratic.
problem
Agile algebra

Age

16 to 18

Challenge level

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

Age

16 to 18

Challenge level

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

Age

16 to 18

Challenge level

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

Age

16 to 18

Challenge level

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.
problem
Exploring cubic functions

Age

14 to 18

Challenge level

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