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There are **23** NRICH Mathematical resources connected to **Polynomial functions and their roots**, you may find related items under Coordinates, functions and graphs.

Problem
Primary curriculum
Secondary curriculum
### Curve Fitter

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

Age 14 to 18

ShortChallenge Level

Problem
Primary curriculum
Secondary curriculum
### Spinners

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

Age 16 to 18

Challenge Level

Article
Primary curriculum
Secondary curriculum
### Telescoping Functions

Take a complicated fraction with the product of five quartics top and bottom and reduce this to a whole number. This is a numerical example involving some clever algebra.

Age 16 to 18

Problem
Primary curriculum
Secondary curriculum
### Exploring Cubic Functions

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

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Mechanical Integration

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Common Divisor

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Root to Poly

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Polynomial Relations

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.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Interpolating Polynomials

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.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Patterns of Inflection

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Symmetrically So

Exploit the symmetry and turn this quartic into a quadratic.

Age 16 to 18

Challenge Level

Article
Primary curriculum
Secondary curriculum
### The Why and How of Substitution

Step back and reflect! This article reviews techniques such as substitution and change of coordinates which enable us to exploit underlying structures to crack problems.

Age 16 to 18

Problem
Primary curriculum
Secondary curriculum
### Agile Algebra

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Fibonacci Fashion

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Poly Fibs

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.

Age 16 to 18

Challenge Level

Article
Primary curriculum
Secondary curriculum
### An Introduction to Galois Theory

This article only skims the surface of Galois theory and should probably be accessible to a 17 or 18 year old school student with a strong interest in mathematics.

Age 16 to 18

Problem
Primary curriculum
Secondary curriculum
### Janine's Conjecture

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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Cubic Spin

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### More Polynomial Equations

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.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Two Cubes

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.]

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Janusz Asked

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?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Real(ly) Numbers

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?

Age 16 to 18

Challenge Level