Standard +0.8 This is a multi-step Further Maths question requiring systematic application of Vieta's formulas and algebraic manipulation of symmetric functions of roots. While the techniques are standard for FP1 (relating sums of roots to coefficients), it requires careful bookkeeping across multiple equations and the insight to use the identity α²+β²+γ² = (α+β+γ)² - 2(αβ+αγ+βγ). The final part requires solving for individual roots given partial information, which adds complexity beyond routine exercises.
5 The cubic equation \(x ^ { 3 } + p x ^ { 2 } + q x + r = 0\), where \(p , q\) and \(r\) are integers, has roots \(\alpha , \beta\) and \(\gamma\), such that
$$\begin{aligned}
\alpha + \beta + \gamma & = 15 \\
\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } & = 83
\end{aligned}$$
Write down the value of \(p\) and find the value of \(q\).
Given that \(\alpha , \beta\) and \(\gamma\) are all real and that \(\alpha \beta + \alpha \gamma = 36\), find \(\alpha\) and hence find the value of \(r\).
5 The cubic equation $x ^ { 3 } + p x ^ { 2 } + q x + r = 0$, where $p , q$ and $r$ are integers, has roots $\alpha , \beta$ and $\gamma$, such that
$$\begin{aligned}
\alpha + \beta + \gamma & = 15 \\
\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } & = 83
\end{aligned}$$
Write down the value of $p$ and find the value of $q$.
Given that $\alpha , \beta$ and $\gamma$ are all real and that $\alpha \beta + \alpha \gamma = 36$, find $\alpha$ and hence find the value of $r$.
\hfill \mbox{\textit{CAIE FP1 2015 Q5 [8]}}