2 Find the cubic equation with roots \(\alpha , \beta\) and \(\gamma\) such that
$$\begin{aligned}
\alpha + \beta + \gamma & = 3 \\
\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } & = 1 \\
\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } & = - 30
\end{aligned}$$
giving your answer in the form \(x ^ { 3 } + p x ^ { 2 } + q x + r = 0\), where \(p , q\) and \(r\) are integers to be found.
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Question 2:
Answer Marks
Guidance
Answer/Working Marks
Guidance
\(2\sum\alpha\beta = 9-1 \Rightarrow \sum\alpha\beta = 4\) M1A1
Use of \(\sum\alpha^3 - 3\alpha\beta\gamma = \sum\alpha\left(\sum\alpha^2 - \sum\alpha\beta\right)\) M1
or \(\left(\sum\alpha\right)^3 = \sum\alpha^3 + 3\sum\alpha\sum\alpha\beta - 3\alpha\beta\gamma\) Correct substitution in formula A1
\(\Rightarrow \alpha\beta\gamma = -7\) A1
Required cubic equation is \(x^3 - 3x^2 + 4x + 7 = 0\) A1\(\checkmark\)
*must see final equation*, [6]
ALT METHOD: \(S_3 - 3S_2 + 4S_1 + 3r = 0\) M1 \(3r = 30 + 3\times1 - 4\times3\) A1 \(r=7\) A1
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## Question 2:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $2\sum\alpha\beta = 9-1 \Rightarrow \sum\alpha\beta = 4$ | M1A1 | |
| Use of $\sum\alpha^3 - 3\alpha\beta\gamma = \sum\alpha\left(\sum\alpha^2 - \sum\alpha\beta\right)$ | M1 | |
| or $\left(\sum\alpha\right)^3 = \sum\alpha^3 + 3\sum\alpha\sum\alpha\beta - 3\alpha\beta\gamma$ | | |
| Correct substitution in formula | A1 | |
| $\Rightarrow \alpha\beta\gamma = -7$ | A1 | |
| Required cubic equation is $x^3 - 3x^2 + 4x + 7 = 0$ | A1$\checkmark$ | *must see final equation*, [6] |
| ALT METHOD: $S_3 - 3S_2 + 4S_1 + 3r = 0$ **M1** $3r = 30 + 3\times1 - 4\times3$ **A1** $r=7$ **A1** | | |
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2 Find the cubic equation with roots $\alpha , \beta$ and $\gamma$ such that
$$\begin{aligned}
\alpha + \beta + \gamma & = 3 \\
\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } & = 1 \\
\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } & = - 30
\end{aligned}$$
giving your answer in the form $x ^ { 3 } + p x ^ { 2 } + q x + r = 0$, where $p , q$ and $r$ are integers to be found.
\hfill \mbox{\textit{CAIE FP1 2016 Q2 [6]}}