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

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 \(\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } }\).

The quartic equation $$2x^4 + Ax^3 - Ax^2 - 5x + 6 = 0$$ where \(A\) is a real constant, has roots \(\alpha\), \(\beta\), \(\gamma\) and \(\delta\)

1. Determine the value of $$\frac{3}{\alpha} + \frac{3}{\beta} + \frac{3}{\gamma} + \frac{3}{\delta}$$ [3]

Given that \(\alpha^2 + \beta^2 + \gamma^2 + \delta^2 = -\frac{3}{4}\)

1. determine the possible values of \(A\) [5]

You are given that the cubic equation \(2x^3 - 3x^2 + x + 4 = 0\) has three roots, \(\alpha\), \(\beta\) and \(\gamma\). By making a suitable substitution to obtain a related cubic equation, determine the value of \(\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma}\). [4]