### There are 14 results

Broad Topics >

Advanced Algebra > Polynomials

##### 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 $1+\sqrt 2+\sqrt 3$.

##### 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. . . .

##### 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.

##### 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. . . .

##### Age 16 to 18

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 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.

##### 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.

##### Age 16 to 18 Challenge Level:

Can you fit polynomials through these points?

##### 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.

##### Age 16 to 18 Challenge Level:

Exploit the symmetry and turn this quartic into a quadratic.

##### 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.

##### 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?

##### 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.

##### Age 16 to 18 Challenge Level:

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