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.]

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.]

Mathematical reasoning & proof. Graph plotters. Arithmetic sequences. Graphs. Polynomial functions and their roots. Creating and manipulating expressions and formulae. Summation of series. Expanding and factorising quadratics. Transformation of functions. Quadratic functions. Making and proving conjectures.