- Given that the roots of the equation \(x^3+px^2+qx+r=0\) are three consecutive terms of an arithmetic progression, show that\[ 2p^3 + 27r = 9pq.\]

Given any cubic equation, then what do the three roots add to? Multiply to?

What if we take the roots in pairs, multiply them together and then add the results?

If we know the three roots are in arithmetic progression, then how can we sensibly write them down?

- Given that the roots of the equation \(x^3+px^2+qx+r=0\) are three consecutive terms of a geometric progression, find a condition that \(p\), \(q\) and \(r\) must satisfy.

How can we use our work above a second time?

If we know the three roots are in geometric progression, then how can we sensibly write them down?