| Exam Board | CAIE |
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| Module | FP1 (Further Pure Mathematics 1) |
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| Year | 2012 |
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| Session | November |
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| Topic | Roots of polynomials |
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7 A cubic equation has roots \(\alpha , \beta\) and \(\gamma\) such that
$$\begin{aligned}
\alpha + \beta + \gamma & = 4
\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } & = 14
\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } & = 34
\end{aligned}$$
Find the value of \(\alpha \beta + \beta \gamma + \gamma \alpha\).
Show that the cubic equation is
$$x ^ { 3 } - 4 x ^ { 2 } + x + 6 = 0$$
and solve this equation.