5 The equation
$$x ^ { 3 } + 5 x + 3 = 0$$
has roots \(\alpha , \beta , \gamma\). Use the substitution \(x = - \frac { 3 } { y }\) to find a cubic equation in \(y\) and show that the roots of this equation are \(\beta \gamma , \gamma \alpha , \alpha \beta\).
Find the exact values of \(\beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 } + \alpha ^ { 2 } \beta ^ { 2 }\) and \(\beta ^ { 3 } \gamma ^ { 3 } + \gamma ^ { 3 } \alpha ^ { 3 } + \alpha ^ { 3 } \beta ^ { 3 }\).
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Shows \(y = -3/x \Rightarrow y^3 - 5y^2 - 9 = 0\) B1
\(y = -3/x \Rightarrow y = a\theta/x\) M1
\(\Rightarrow y = [\beta y, \gamma a\beta] \text{ when } x = a, \beta, \gamma, \text{ respectively}\) M1A1
OR for previous 3 marks:\([\beta y + \gamma a + a\beta = 5\) B1
\(a[^2\beta y + a\beta^2y + a\beta\gamma^2 = a\beta\gamma(a + \beta + \gamma) = a\beta \times 0 = 0\) B1
\(\beta\gamma a a\beta = (a\beta\gamma)^2 = 9\) B1
\(\Sigma a[\beta]^2 = 25 - (2 \times 0) = 25\) M1A1
\(\Sigma a[\beta]^3 - 5\Sigma a^2[\beta]^2 - 27 = 0\) M1A1
\(\Rightarrow ... \Rightarrow \Sigma a[\beta]^3 = 152\) A1
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Shows $y = -3/x \Rightarrow y^3 - 5y^2 - 9 = 0$ | B1 |
$y = -3/x \Rightarrow y = a\theta/x$ | M1 |
$\Rightarrow y = [\beta y, \gamma a\beta] \text{ when } x = a, \beta, \gamma, \text{ respectively}$ | M1A1 |
**OR** for previous 3 marks: | |
$[\beta y + \gamma a + a\beta = 5$ | B1 |
$a[^2\beta y + a\beta^2y + a\beta\gamma^2 = a\beta\gamma(a + \beta + \gamma) = a\beta \times 0 = 0$ | B1 |
$\beta\gamma a a\beta = (a\beta\gamma)^2 = 9$ | B1 |
$\Sigma a[\beta]^2 = 25 - (2 \times 0) = 25$ | M1A1 |
$\Sigma a[\beta]^3 - 5\Sigma a^2[\beta]^2 - 27 = 0$ | M1A1 |
$\Rightarrow ... \Rightarrow \Sigma a[\beta]^3 = 152$ | A1 |
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5 The equation
$$x ^ { 3 } + 5 x + 3 = 0$$
has roots $\alpha , \beta , \gamma$. Use the substitution $x = - \frac { 3 } { y }$ to find a cubic equation in $y$ and show that the roots of this equation are $\beta \gamma , \gamma \alpha , \alpha \beta$.
Find the exact values of $\beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 } + \alpha ^ { 2 } \beta ^ { 2 }$ and $\beta ^ { 3 } \gamma ^ { 3 } + \gamma ^ { 3 } \alpha ^ { 3 } + \alpha ^ { 3 } \beta ^ { 3 }$.
\hfill \mbox{\textit{CAIE FP1 2009 Q5 [9]}}