Equation with nonlinearly transformed roots

3 In this question you must show detailed reasoning. The roots of the equation \(5 x ^ { 3 } - 3 x ^ { 2 } - 2 x + 9 = 0\) are \(\alpha , \beta\) and \(\gamma\).
Find a cubic equation with integer coefficients whose roots are \(\alpha \beta , \beta \gamma\) and \(\gamma \alpha\).

3 In this question you must show detailed reasoning. The roots of the equation \(4 x ^ { 3 } + 6 x ^ { 2 } - 3 x + 9 = 0\) are \(\alpha , \beta\) and \(\gamma\). Find a cubic equation with integer coefficients whose roots are \(\alpha + \beta , \beta + \gamma\) and \(\gamma + \alpha\).

4 The cubic equation $$z ^ { 3 } + \mathrm { i } z ^ { 2 } + 3 z - ( 1 + \mathrm { i } ) = 0$$ has roots \(\alpha , \beta\) and \(\gamma\).

1. Write down the value of:
   1. \(\alpha + \beta + \gamma\);
   2. \(\alpha \beta + \beta \gamma + \gamma \alpha\);
   3. \(\alpha \beta \gamma\).
2. Find the value of:
   1. \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\);
   2. \(\alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 }\);
   3. \(\alpha ^ { 2 } \beta ^ { 2 } \gamma ^ { 2 }\).
3. Hence write down a cubic equation whose roots are \(\alpha ^ { 2 } , \beta ^ { 2 }\) and \(\gamma ^ { 2 }\).

5 In this question you must show detailed reasoning. You are given that \(\alpha , \beta\) and \(\gamma\) are the roots of the equation \(5 x ^ { 3 } - 2 x ^ { 2 } + 3 x + 1 = 0\).

1. Find the value of \(\alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 }\).
2. Find a cubic equation whose roots are \(\alpha ^ { 2 } , \beta ^ { 2 }\) and \(\gamma ^ { 2 }\) giving your answer in the form \(a x ^ { 3 } + b x ^ { 2 } + c x + d = 0\) where \(a , b , c\) and \(d\) are integers.

3 In this question you must show detailed reasoning.
The cubic equation \(5 x ^ { 3 } + 3 x ^ { 2 } - 4 x + 7 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
Find a cubic equation with integer coefficients whose roots are \(\alpha + \beta , \beta + \gamma\) and \(\gamma + \alpha\).

1 In this question you must show detailed reasoning.
The roots of the equation \(3 x ^ { 3 } - 2 x ^ { 2 } - 5 x - 4 = 0\) are \(\alpha , \beta\) and \(\gamma\).

1. Find a cubic equation with integer coefficients whose roots are \(\alpha ^ { 2 } , \beta ^ { 2 }\) and \(\gamma ^ { 2 }\).
2. Find the exact value of \(\frac { \alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 } } { \alpha \beta \gamma }\).

The quartic equation \(x^4 + 2x^3 - 1 = 0\) has roots \(\alpha, \beta, \gamma, \delta\).

1. Find a quartic equation whose roots are \(\alpha^4, \beta^4, \gamma^4, \delta^4\) and state the value of \(\alpha^4 + \beta^4 + \gamma^4 + \delta^4\). [5]
2. Find the value of \(\alpha^5 + \beta^5 + \gamma^5 + \delta^5\). [3]
3. Find the value of \(\alpha^8 + \beta^8 + \gamma^8 + \delta^8\). [2]

The equation \(x^3 + 2x^2 + x + 7 = 0\) has roots \(\alpha\), \(\beta\), \(\gamma\).

1. Use the relation \(x^2 = -7y\) to show that the equation $$49y^3 + 14y^2 - 27y + 7 = 0$$ has roots \(\frac{\alpha}{\beta \gamma}\), \(\frac{\beta}{\gamma \alpha}\), \(\frac{\gamma}{\alpha \beta}\). [4]
2. Show that \(\frac{\alpha^2}{\beta^2 \gamma^2} + \frac{\beta^2}{\gamma^2 \alpha^2} + \frac{\gamma^2}{\alpha^2 \beta^2} = \frac{58}{49}\). [3]
3. Find the exact value of \(\frac{\alpha^2}{\beta^3 \gamma^3} + \frac{\beta^2}{\gamma^3 \alpha^3} + \frac{\gamma^2}{\alpha^3 \beta^3}\). [2]

The quadratic equation $$2x^2 + 8x + 1 = 0$$ has roots \(\alpha\) and \(\beta\).

1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha\beta\). [2 marks]
2. 1. Find the value of \(\alpha^2 + \beta^2\). [2 marks]
   2. Hence, or otherwise, show that \(\alpha^4 + \beta^4 = \frac{449}{2}\). [2 marks]
3. Find a quadratic equation, with integer coefficients, which has roots $$2\alpha^4 + \frac{1}{\beta^2} \text{ and } 2\beta^4 + \frac{1}{\alpha^2}$$ [5 marks]

The quadratic equation \(x^2 + 2kx + k = 0\), where \(k\) is a non-zero constant, has roots \(\alpha\) and \(\beta\). Find a quadratic equation with roots \(\frac{\alpha + \beta}{\alpha}\) and \(\frac{\alpha + \beta}{\beta}\). [7]

In this question you must show detailed reasoning. You are given that \(\alpha\), \(\beta\) and \(\gamma\) are the roots of the equation \(5x^3 - 2x^2 + 3x + 1 = 0\).

1. Find the value of \(\alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2\). [5]
2. Find a cubic equation whose roots are \(\alpha^2\), \(\beta^2\) and \(\gamma^2\) giving your answer in the form \(ax^3 + bx^2 + cx + d = 0\) where \(a\), \(b\), \(c\) and \(d\) are integers. [4]

**In this question you must show detailed reasoning.** The equation \(x^2 + 2x + 5 = 0\) has roots \(\alpha\) and \(\beta\). The equation \(x^2 + px + q = 0\) has roots \(\alpha^2\) and \(\beta^2\). Find the values of \(p\) and \(q\). [3]

The cubic equation \(4x^3 - 12x^2 + 9x - 16 = 0\) has roots \(r_1\), \(r_2\) and \(r_3\). A second cubic equation, with integer coefficients, has roots \(R_1 = \frac{r_2 + r_3}{r_1}\), \(R_2 = \frac{r_3 + r_1}{r_2}\) and \(R_3 = \frac{r_1 + r_2}{r_3}\).

1. Show that \(1 + R_1 = \frac{3}{r_1}\) and write down the corresponding results for the other roots. [2]
2. Using a substitution based on this result, or otherwise, find this second cubic equation. [6]