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# Polynomial Relations

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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.

Let $p(x) = x^2 + 2x$ and $q(x) = x^2 + x + 1$. Then, using a method which does not depend on knowing the answer, show that the relationship between the polynomials is:

\[ p^2 - 2pq + q^2 + 3p - 4q + 3 = 0 \]

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?

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?

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.