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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.

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Mechanical Integration

To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.

Janusz Asked

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2
Start with the linear polynomial: $y = -3x + 9$. The $x$-coefficient, the root and the intercept are -3, 3 and 9 respectively, and these are in arithmetic progression. Are there any other linear polynomials that enjoy this property?

What about quadratic polynomials? That is, if the polynomial \[y = ax^2 + bx + c\] has roots $r_1$ and $r_2,$ can $a$, $r_1$, $b$, $r_2$ and $c$ be in arithmetic progression?

[The idea for this problem came from Janusz Kowalski of the Kreator Project.]