### 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?

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?

### 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.

# More Polynomial Equations

##### Age 16 to 18Challenge Level

Here you need to start with $$\begin{eqnarray} a(n) &= &1 + 2 + 3 + ... + n &= & {1\over 2}n(n+1)\\ b(n) &= &1^2 + 2^2 + 3^2 + ... + n^2 &= & {1\over 6}n(n+1)(2n + 1)\\ c(n) &= &1^3 + 2^3 + 3^3 + ... + n^3 &= & {1\over 4}n^2(n+1)^2. \end{eqnarray}$$

These are well known results found in many text books and you will find proofs in some problems on this website. See Picture Story and Natural Sum.