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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?

Janusz Asked

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 18
Challenge 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 $q=x^2$.