Reciprocal sum of roots

1. The quadratic equation

$$3 x ^ { 2 } - 5 x + 1 = 0$$ has roots \(\alpha\) and \(\beta\).
Without solving the quadratic equation, find the exact value of $$\frac { \alpha } { \beta } + \frac { \beta } { \alpha }$$ Count coution
\(\_\_\_\_\) T

9

1. Show that \(( \alpha \beta + \beta \gamma + \gamma \alpha ) ^ { 2 } \equiv \alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 } + 2 \alpha \beta \gamma ( \alpha + \beta + \gamma )\).
2. It is given that \(\alpha , \beta\) and \(\gamma\) are the roots of the cubic equation \(x ^ { 3 } + p x ^ { 2 } - 4 x + 3 = 0\), where \(p\) is a constant. Find the value of \(\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } } + \frac { 1 } { \gamma ^ { 2 } }\) in terms of \(p\).

4 The roots of the quadratic equation \(x ^ { 2 } + x - 8 = 0\) are \(p\) and \(q\). Find the value of \(p + q + \frac { 1 } { p } + \frac { 1 } { q }\).

5 In this question you must show detailed reasoning. The roots of the equation \(5 x ^ { 2 } - 3 x + 12 = 0\) are \(\alpha\) and \(\beta\). By considering the symmetric functions of the roots, \(\alpha + \beta\) and \(\alpha \beta\), determine the exact value of \(\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } }\).