Polynomial functions and their roots

Given any two polynomials in a single variable it is always possible to eliminate the variable and obtain a formula showing the relationship between the two polynomials. Try this one.

Find the polynomial p(x) with integer coefficients such that one solution of the equation p(x)=0 is $1+\sqrt 2+\sqrt 3$.

Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.

To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.

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

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.