### 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?

### 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.

# Polynomial Relations

##### Age 16 to 18Challenge Level

Why do this problem?
It gives practice in manipulation of polynomials.

Possible approach
An easy lesson starter!

Key question
What is $p(x)-q(x)$?

Possible extension
Learners can make up their own probems by writing down two polynomials in $x$ and then eliminating $x$ between the expressions. They might be asked to make up such a problem and exchange problems with their partner. Then they can compare and check results in pairs.

Possible support
Try a simpler example such as: find the formula relating

$p$ and $q$ where $p=x+3$ and $q=x^2$.