2 The roots of the equation
$$x ^ { 3 } + p x ^ { 2 } + q x + r = 0$$
are \(\frac { \beta } { k } , \beta , k \beta\), where \(p , q , r , k\) and \(\beta\) are non-zero real constants. Show that \(\beta = - \frac { q } { p }\).
Deduce that \(r p ^ { 3 } = q ^ { 3 }\).
2 The roots of the equation
$$x ^ { 3 } + p x ^ { 2 } + q x + r = 0$$
are $\frac { \beta } { k } , \beta , k \beta$, where $p , q , r , k$ and $\beta$ are non-zero real constants. Show that $\beta = - \frac { q } { p }$.
Deduce that $r p ^ { 3 } = q ^ { 3 }$.
\hfill \mbox{\textit{CAIE FP1 2011 Q2}}