In this question, the numbers \(a\), \(b\) and \(c\) may be complex.
Let \(p\), \(q\) and \(r\) be real numbers. Given that there are numbers \(a\) and \(b\) such that
\[
a + b = p, \quad a^2 + b^2 = q \quad \text{and} \quad a^3 + b^3 = r, \qquad (*)
\]
show that \(3pq - p^3 = 2r\).
Conversely, you are given that the real numbers \(p\), \(q\) and \(r\) satisfy \(3pq - p^3 = 2r\). By considering the equation \(2x^2 - 2px + (p^2 - q) = 0\), show that there exist numbers \(a\) and \(b\) such that the three equations \((*)\) hold.
Let \(s\), \(t\), \(u\) and \(v\) be real numbers. Given that there are distinct numbers \(a\), \(b\) and \(c\) such that
\[
a + b + c = s, \quad a^2 + b^2 + c^2 = t, \quad a^3 + b^3 + c^3 = u \quad \text{and} \quad abc = v,
\]
show, using part~(i), that \(c\) is a root of the equation
\[
6x^3 - 6sx^2 + 3(s^2 - t)x + 3st - s^3 - 2u = 0
\]
and write down the other two roots.
Deduce that \(s^3 - 3st + 2u = 6v\).
Find numbers \(a\), \(b\) and \(c\) such that
\[
a + b + c = 3, \quad a^2 + b^2 + c^2 = 1, \quad a^3 + b^3 + c^3 = -3 \quad \text{and} \quad abc = 2, \qquad (**)
\]
and verify that your solution satisfies the four equations \((**)\).