Roots with given sum conditions

3 The equation \(x ^ { 3 } + p x ^ { 2 } + q x + 3 = 0\) has roots \(\alpha , \beta\) and \(\gamma\), where $$\begin{gathered} \alpha + \beta + \gamma = 4 \\ \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 6 \end{gathered}$$ Find \(p\) and \(q\).

3 The equation \(2 \mathrm { x } ^ { 3 } + \mathrm { px } ^ { 2 } + \mathrm { qx } + \mathrm { r } = 0\) has a root at \(x = 4\). The sum of the roots is 6 and the product of the roots is - 10 . Find \(p , q\) and \(r\).

5 The cubic equation \(x ^ { 3 } + p x ^ { 2 } + q x + r = 0\), where \(p , q\) and \(r\) are integers, has roots \(\alpha , \beta\) and \(\gamma\), such that $$\begin{aligned} \alpha + \beta + \gamma & = 15 \\ \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } & = 83 \end{aligned}$$

1. Write down the value of \(p\) and find the value of \(q\).
2. Given that \(\alpha , \beta\) and \(\gamma\) are all real and that \(\alpha \beta + \alpha \gamma = 36\), find \(\alpha\) and hence find the value of \(r\). [5]

5 The cubic equation \(x ^ { 3 } + p x ^ { 2 } + q x + r = 0\), where \(p , q\) and \(r\) are integers, has roots \(\alpha , \beta\) and \(\gamma\), such that $$\begin{aligned} \alpha + \beta + \gamma & = 15 \\ \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } & = 83 \end{aligned}$$ Write down the value of \(p\) and find the value of \(q\). Given that \(\alpha , \beta\) and \(\gamma\) are all real and that \(\alpha \beta + \alpha \gamma = 36\), find \(\alpha\) and hence find the value of \(r\).

7 The numbers \(\alpha , \beta\) and \(\gamma\) satisfy the equations $$\begin{aligned} & \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = - 10 - 12 \mathrm { i } \\ & \alpha \beta + \beta \gamma + \gamma \alpha = 5 + 6 \mathrm { i } \end{aligned}$$

1. Show that \(\alpha + \beta + \gamma = 0\).
2. The numbers \(\alpha , \beta\) and \(\gamma\) are also the roots of the equation $$z ^ { 3 } + p z ^ { 2 } + q z + r = 0$$ Write down the value of \(p\) and the value of \(q\).
3. It is also given that \(\alpha = 3 \mathrm { i }\).
   1. Find the value of \(r\).
   2. Show that \(\beta\) and \(\gamma\) are the roots of the equation $$z ^ { 2 } + 3 \mathrm { i } z - 4 + 6 \mathrm { i } = 0$$
   3. Given that \(\beta\) is real, find the values of \(\beta\) and \(\gamma\).

3 The cubic equation $$z ^ { 3 } + q z + ( 18 - 12 i ) = 0$$ where \(q\) is a complex number, has roots \(\alpha , \beta\) and \(\gamma\).

1. Write down the value of:
   1. \(\alpha \beta \gamma\);
   2. \(\alpha + \beta + \gamma\).
2. Given that \(\beta + \gamma = 2\), find the value of:
   1. \(\alpha\);
   2. \(\quad \beta \gamma\);
   3. \(q\).
3. Given that \(\beta\) is of the form \(k \mathrm { i }\), where \(k\) is real, find \(\beta\) and \(\gamma\).

4 The roots of the cubic equation $$z ^ { 3 } - 2 z ^ { 2 } + p z + 10 = 0$$ are \(\alpha , \beta\) and \(\gamma\).
It is given that \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = - 4\).

1. Write down the value of \(\alpha + \beta + \gamma\).
2. 1. Explain why \(\alpha ^ { 3 } - 2 \alpha ^ { 2 } + p \alpha + 10 = 0\).
   2. Hence show that $$\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = p + 13$$
   3. Deduce that \(p = - 3\).
3. 1. Find the real root \(\alpha\) of the cubic equation \(z ^ { 3 } - 2 z ^ { 2 } - 3 z + 10 = 0\).
   2. Find the values of \(\beta\) and \(\gamma\).

1. The cubic equation

$$4 x ^ { 3 } + p x ^ { 2 } - 14 x + q = 0$$ where \(p\) and \(q\) are real positive constants, has roots \(\alpha , \beta\) and \(\gamma\) Given that \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 16\)

1. show that \(p = 12\) Given that \(\frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma } = \frac { 14 } { 3 }\)
2. determine the value of \(q\) Without solving the cubic equation,
3. determine the value of \(( \alpha - 1 ) ( \beta - 1 ) ( \gamma - 1 )\)

5 The cubic equation $$z ^ { 3 } - 4 \mathrm { i } z ^ { 2 } + q z - ( 4 - 2 \mathrm { i } ) = 0$$ where \(q\) is a complex number, has roots \(\alpha , \beta\) and \(\gamma\).

1. Write down the value of:
   1. \(\alpha + \beta + \gamma\);
   2. \(\alpha \beta \gamma\).
2. Given that \(\alpha = \beta + \gamma\), show that:
   1. \(\alpha = 2 \mathrm { i }\);
   2. \(\quad \beta \gamma = - ( 1 + 2 \mathrm { i } )\);
   3. \(\quad q = - ( 5 + 2 \mathrm { i } )\).
3. Show that \(\beta\) and \(\gamma\) are the roots of the equation $$z ^ { 2 } - 2 \mathrm { i } z - ( 1 + 2 \mathrm { i } ) = 0$$
4. Given that \(\beta\) is real, find \(\beta\) and \(\gamma\).

The cubic equation \(x^3 + px^2 + qx + r = 0\), where \(p\), \(q\) and \(r\) are integers, has roots \(\alpha\), \(\beta\) and \(\gamma\), such that $$\alpha + \beta + \gamma = 15,$$ $$\alpha^2 + \beta^2 + \gamma^2 = 83.$$ Write down the value of \(p\) and find the value of \(q\). [3] Given that \(\alpha\), \(\beta\) and \(\gamma\) are all real and that \(\alpha\beta + \alpha\gamma = 36\), find \(\alpha\) and hence find the value of \(r\). [5]

The roots of the equation $$z^3 - 5z^2 + kz - 4 = 0$$ are \(\alpha\), \(\beta\) and \(\gamma\).

1. 1. Write down the value of \(\alpha + \beta + \gamma\) and the value of \(\alpha\beta\gamma\). [2 marks]
   2. Hence find the value of \(\alpha^2\beta\gamma + \alpha\beta^2\gamma + \alpha\beta\gamma^2\). [2 marks]
2. The value of \(\alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2\) is \(-4\).
   1. Explain why \(\alpha\), \(\beta\) and \(\gamma\) cannot all be real. [1 mark]
   2. By considering \((\alpha\beta + \beta\gamma + \gamma\alpha)^2\), find the possible values of \(k\). [4 marks]

\(\alpha\), \(\beta\) and \(\gamma\) are the real roots of the cubic equation $$x^3 + mx^2 + nx + 2 = 0$$ By considering \((\alpha - \beta)^2 + (\gamma - \alpha)^2 + (\beta - \gamma)^2\), prove that $$m^2 \geq 3n$$ [4 marks]