3 The cubic equation \(x ^ { 3 } - 2 x ^ { 2 } - 3 x + 4 = 0\) has roots \(\alpha , \beta , \gamma\). Given that \(c = \alpha + \beta + \gamma\), state the value of \(c\).
Use the substitution \(y = c - x\) to find a cubic equation whose roots are \(\alpha + \beta , \beta + \gamma , \gamma + \alpha\).
Find a cubic equation whose roots are \(\frac { 1 } { \alpha + \beta } , \frac { 1 } { \beta + \gamma } , \frac { 1 } { \gamma + \alpha }\).
Hence evaluate \(\frac { 1 } { ( \alpha + \beta ) ^ { 2 } } + \frac { 1 } { ( \beta + \gamma ) ^ { 2 } } + \frac { 1 } { ( \gamma + \alpha ) ^ { 2 } }\).
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3 The cubic equation $x ^ { 3 } - 2 x ^ { 2 } - 3 x + 4 = 0$ has roots $\alpha , \beta , \gamma$. Given that $c = \alpha + \beta + \gamma$, state the value of $c$.
Use the substitution $y = c - x$ to find a cubic equation whose roots are $\alpha + \beta , \beta + \gamma , \gamma + \alpha$.
Find a cubic equation whose roots are $\frac { 1 } { \alpha + \beta } , \frac { 1 } { \beta + \gamma } , \frac { 1 } { \gamma + \alpha }$.
Hence evaluate $\frac { 1 } { ( \alpha + \beta ) ^ { 2 } } + \frac { 1 } { ( \beta + \gamma ) ^ { 2 } } + \frac { 1 } { ( \gamma + \alpha ) ^ { 2 } }$.
\hfill \mbox{\textit{CAIE FP1 2013 Q3}}