Polynomial functions and their roots

problem

More Polynomial Equations

Age

16 to 18

Challenge level

Find relationships between the polynomials a, b and c which are polynomials in n giving the sums of the first n natural numbers, squares and cubes respectively.
problem

Cubic Spin

Age

16 to 18

Challenge level

Prove that the graph of f(x) = x^3 - 6x^2 +9x +1 has rotational symmetry. Do graphs of all cubics have rotational symmetry?
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

Janine's Conjecture

Age

14 to 16

Challenge level

Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. Does this always work? Can you prove or disprove this conjecture?
problem

Agile Algebra

Age

16 to 18

Challenge level

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

Polynomial Interpolation

Age

16 to 18

Challenge level

Can you fit polynomials through these points?
problem

Real(ly) Numbers

Age

16 to 18

Challenge level

If x, y and z are real numbers such that: x + y + z = 5 and xy + yz + zx = 3. What is the largest value that any of the numbers can have?
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.