Interpolating Polynomials
To find four points that a quadratic couldn't possibly fit, remember that quadratics only have one turning point.
To find the quadratic that fits three points, make sure you understand how you can add and subtract graphs, and what happens to the result. Don't try to fit all three points at once – fit two and then “fix” your line to fit the third.
Uniqueness: The Factor Theorem states that if $p$ is a polynomial and $p(a) = 0$, then there is a polynomial $q$ such that $p(x) = (x-a)q(x)$. What does this mean about the degrees of $p$ and $q$?
Finally, to prove two polynomials are equal, try proving their difference is zero.
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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?
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.