| Exam Board | CAIE |
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| Module | FP1 (Further Pure Mathematics 1) |
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| Year | 2011 |
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| Session | November |
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| Topic | Roots of polynomials |
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3 The equation
$$x ^ { 3 } + 5 x ^ { 2 } - 3 x - 15 = 0$$
has roots \(\alpha , \beta , \gamma\). Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\).
Hence show that the matrix \(\left( \begin{array} { c c c } 1 & \alpha & \beta
\alpha & 1 & \gamma
\beta & \gamma & 1 \end{array} \right)\) is singular.