Standard +0.8 This is a Further Maths question requiring systematic application of Vieta's formulas with algebraically related roots. Students must manipulate the sum and product of roots (β/k + β + kβ = -p and β³ = -r) to eliminate k and establish two non-trivial relationships. While the techniques are standard, the algebraic manipulation across multiple steps and the need to strategically combine equations elevates this above routine problems.
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 [6]}}