Standard +0.8 This is a Further Maths question requiring insight into root relationships and symmetric functions. Students must recognize that if α and 1/α are roots, then using Vieta's formulas with the third root β leads to the product of roots being β = q, then cleverly manipulating the sum of products to derive the required identity. It requires more algebraic sophistication than standard A-level but is a well-defined proof with clear structure.
1 The equation \(x ^ { 3 } + p x + q = 0\), where \(p\) and \(q\) are constants, with \(q \neq 0\), has one root which is the reciprocal of another root. Prove that \(p + q ^ { 2 } = 1\).
1 The equation $x ^ { 3 } + p x + q = 0$, where $p$ and $q$ are constants, with $q \neq 0$, has one root which is the reciprocal of another root. Prove that $p + q ^ { 2 } = 1$.
\hfill \mbox{\textit{CAIE FP1 2014 Q1 [5]}}