Expanding and factorising quadratics

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

Leftovers

Age

14 to 16

Challenge level

Weekly Problem 26 - 2008
If $n$ is a positive integer, how many different values for the remainder are obtained when $n^2$ is divided by $n+4$?
problem

Geometric Parabola

Age

14 to 16

Challenge level

Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.
problem

Unit Interval

Age

16 to 18

Challenge level

Can you prove our inequality holds for all values of x and y between 0 and 1?
problem

Powerful Factors

Age

16 to 18

Challenge level

Use the fact that: x²-y² = (x-y)(x+y) and x³+y³ = (x+y) (x²-xy+y²) to find the highest power of 2 and the highest power of 3 which divide 5^{36}-1.
problem

Spot the Difference

Age

16 to 18

Challenge level

If you plot these graphs they may look the same, but are they?
problem

Polar Flower

Age

16 to 18

Challenge level

This polar equation is a quadratic. Plot the graph given by each factor to draw the flower.
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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.