Polynomial functions and their roots

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Favourite

Spinners

Age

16 to 18

Challenge level

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

Root to Poly

Age

14 to 16

Challenge level

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

Common Divisor

Age

14 to 16

Challenge level

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

Exploring Cubic Functions

Age

14 to 18

Challenge level

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

Curve Fitter

Age

14 to 18

Challenge level

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

Age

16 to 18

Challenge level

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.
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Favourite

Mechanical Integration

Age

16 to 18

Challenge level

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

Age

14 to 16

Challenge level

Two cubes, each with integral side lengths, have a combined volume equal to the total of the lengths of their edges. How big are the cubes? [If you find a result by 'trial and error' you'll need to prove you have found all possible solutions.]
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Symmetrically So

Age

16 to 18

Challenge level

Exploit the symmetry and turn this quartic into a quadratic.
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Janusz Asked

Age

16 to 18

Challenge level

In y = ax +b when are a, -b/a, b in arithmetic progression. The polynomial y = ax^2 + bx + c has roots r1 and r2. Can a, r1, b, r2 and c be in arithmetic progression?