4.05a Roots and coefficients: symmetric functions

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

5

1. 1. Calculate \(( 2 + \mathrm { i } \sqrt { 5 } ) ( \sqrt { 5 } - \mathrm { i } )\).
   2. Hence verify that \(\sqrt { 5 } - \mathrm { i }\) is a root of the equation $$( 2 + \mathrm { i } \sqrt { 5 } ) z = 3 z ^ { * }$$ where \(z ^ { * }\) is the conjugate of \(z\).
2. The quadratic equation $$x ^ { 2 } + p x + q = 0$$ in which the coefficients \(p\) and \(q\) are real, has a complex root \(\sqrt { 5 } - \mathrm { i }\).
   1. Write down the other root of the equation.
   2. Find the sum and product of the two roots of the equation.
   3. Hence state the values of \(p\) and \(q\).

3 The quadratic equation $$2 x ^ { 2 } + 4 x + 3 = 0$$ has roots \(\alpha\) and \(\beta\).

1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
2. Show that \(\alpha ^ { 2 } + \beta ^ { 2 } = 1\).
3. Find the value of \(\alpha ^ { 4 } + \beta ^ { 4 }\).

1 The quadratic equation \(x ^ { 2 } - 6 x + 18 = 0\) has roots \(\alpha\) and \(\beta\).

1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
2. Find a quadratic equation, with integer coefficients, which has roots \(\alpha ^ { 2 }\) and \(\beta ^ { 2 }\).
3. Hence find the values of \(\alpha ^ { 2 }\) and \(\beta ^ { 2 }\).

5 The roots of the quadratic equation $$x ^ { 2 } + 2 x - 5 = 0$$ are \(\alpha\) and \(\beta\).

1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha \beta\).
2. Calculate the value of \(\alpha ^ { 2 } + \beta ^ { 2 }\).
3. Find a quadratic equation which has roots \(\alpha ^ { 3 } \beta + 1\) and \(\alpha \beta ^ { 3 } + 1\).

4 The quadratic equation $$2 x ^ { 2 } - x + 4 = 0$$ has roots \(\alpha\) and \(\beta\).

1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
2. Show that \(\frac { 1 } { \alpha } + \frac { 1 } { \beta } = \frac { 1 } { 4 }\).
3. Find a quadratic equation with integer coefficients such that the roots of the equation are $$\frac { 4 } { \alpha } \text { and } \frac { 4 } { \beta }$$ (3 marks)

1 The equation $$x ^ { 2 } + x + 5 = 0$$ has roots \(\alpha\) and \(\beta\).

1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
2. Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 }\).
3. Show that \(\frac { \alpha } { \beta } + \frac { \beta } { \alpha } = - \frac { 9 } { 5 }\).
4. Find a quadratic equation, with integer coefficients, which has roots \(\frac { \alpha } { \beta }\) and \(\frac { \beta } { \alpha }\).

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

1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
2. Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 }\).
3. Find a quadratic equation which has roots \(4 \alpha ^ { 2 }\) and \(4 \beta ^ { 2 }\). Give your answer in the form \(x ^ { 2 } + p x + q = 0\), where \(p\) and \(q\) are integers.

8 The quadratic equation $$x ^ { 2 } - 4 x + 10 = 0$$ has roots \(\alpha\) and \(\beta\).

1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
2. Show that \(\frac { 1 } { \alpha } + \frac { 1 } { \beta } = \frac { 2 } { 5 }\).
3. Find a quadratic equation, with integer coefficients, which has roots \(\alpha + \frac { 2 } { \beta }\) and \(\beta + \frac { 2 } { \alpha }\).

2 The equation $$4 x ^ { 2 } + 6 x + 3 = 0$$ has roots \(\alpha\) and \(\beta\).

1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
2. Show that \(\alpha ^ { 2 } + \beta ^ { 2 } = \frac { 3 } { 4 }\).
3. Find an equation, with integer coefficients, which has roots $$3 \alpha - \beta \text { and } 3 \beta - \alpha$$

1 The quadratic equation $$5 x ^ { 2 } - 7 x + 1 = 0$$ has roots \(\alpha\) and \(\beta\).

1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
2. Show that \(\frac { \alpha } { \beta } + \frac { \beta } { \alpha } = \frac { 39 } { 5 }\).
3. Find a quadratic equation, with integer coefficients, which has roots $$\alpha + \frac { 1 } { \alpha } \quad \text { and } \quad \beta + \frac { 1 } { \beta }$$ (5 marks)

6 The equation $$2 x ^ { 2 } + 3 x - 6 = 0$$ has roots \(\alpha\) and \(\beta\).

1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha \beta\).
2. Hence show that \(\alpha ^ { 3 } + \beta ^ { 3 } = - \frac { 135 } { 8 }\).
3. Find a quadratic equation, with integer coefficients, whose roots are \(\alpha + \frac { \alpha } { \beta ^ { 2 } }\) and \(\beta + \frac { \beta } { \alpha ^ { 2 } }\).

1 The quadratic equation \(2 x ^ { 2 } + 6 x + 7 = 0\) has roots \(\alpha\) and \(\beta\).

1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha \beta\).
2. Find a quadratic equation, with integer coefficients, which has roots \(\alpha ^ { 2 } - 1\) and \(\beta ^ { 2 } - 1\).
3. Hence find the values of \(\alpha ^ { 2 }\) and \(\beta ^ { 2 }\).

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 \(\gamma\).

4 The cubic equation $$z ^ { 3 } + p z + q = 0$$ has roots \(\alpha , \beta\) and \(\gamma\).

1. 1. Write down the value of \(\alpha + \beta + \gamma\).
   2. Express \(\alpha \beta \gamma\) in terms of \(q\).
2. Show that $$\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = 3 \alpha \beta \gamma$$
3. Given that \(\alpha = 4 + 7 \mathrm { i }\) and that \(p\) and \(q\) are real, find the values of:
   1. \(\beta\) and \(\gamma\);
   2. \(p\) and \(q\).
4. Find a cubic equation with integer coefficients which has roots \(\frac { 1 } { \alpha } , \frac { 1 } { \beta }\) and \(\frac { 1 } { \gamma }\).

5 The cubic equation $$z ^ { 3 } + p z ^ { 2 } + q z + 37 - 36 \mathrm { i } = 0$$ where \(p\) and \(q\) are constants, has three complex roots, \(\alpha , \beta\) and \(\gamma\). It is given that \(\beta = - 2 + 3 \mathrm { i }\) and \(\gamma = 1 + 2 \mathrm { i }\).

1. 1. Write down the value of \(\alpha \beta \gamma\).
   2. Hence show that \(( 8 + \mathrm { i } ) \alpha = 37 - 36 \mathrm { i }\).
   3. Hence find \(\alpha\), giving your answer in the form \(m + n \mathrm { i }\), where \(m\) and \(n\) are integers.
2. Find the value of \(p\).
3. Find the value of the complex number \(q\).

8

1. 1. Use de Moivre's theorem to show that $$\cos 4 \theta = \cos ^ { 4 } \theta - 6 \cos ^ { 2 } \theta \sin ^ { 2 } \theta + \sin ^ { 4 } \theta$$ and find a similar expression for \(\sin 4 \theta\).
   2. Deduce that $$\tan 4 \theta = \frac { 4 \tan \theta - 4 \tan ^ { 3 } \theta } { 1 - 6 \tan ^ { 2 } \theta + \tan ^ { 4 } \theta }$$
2. Explain why \(t = \tan \frac { \pi } { 16 }\) is a root of the equation $$t ^ { 4 } + 4 t ^ { 3 } - 6 t ^ { 2 } - 4 t + 1 = 0$$ and write down the three other roots in trigonometric form.
3. Hence show that $$\tan ^ { 2 } \frac { \pi } { 16 } + \tan ^ { 2 } \frac { 3 \pi } { 16 } + \tan ^ { 2 } \frac { 5 \pi } { 16 } + \tan ^ { 2 } \frac { 7 \pi } { 16 } = 28$$

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

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

2 In this question you must show detailed reasoning.
The equation \(\mathrm { x } ^ { 2 } - \mathrm { kx } + 2 \mathrm { k } = 0\), where \(k\) is a non-zero constant, has roots \(\alpha\) and \(\beta\).
Find \(\frac { \alpha } { \beta } + \frac { \beta } { \alpha }\) in terms of \(k\), simplifying your answer.

10 The equation \(4 x ^ { 4 } + 16 x ^ { 3 } + a x ^ { 2 } + b x + 6 = 0\),
where \(a\) and \(b\) are real, has roots \(\alpha , \frac { 2 } { \alpha } , \beta\) and \(3 \beta\).

1. Given that \(\beta < 0\), determine all 4 roots.
2. Determine the values of \(a\) and \(b\).

10 The equation \(\mathrm { x } ^ { 3 } - 4 \mathrm { x } ^ { 2 } + 7 \mathrm { x } + \mathrm { c } = 0\), where \(c\) is a constant, has roots \(\alpha , \beta\) and \(\alpha + \beta\).

1. Determine the roots of the equation.
2. Find c.

3 The equation \(2 x ^ { 3 } - 2 x ^ { 2 } + 8 x - 15 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
Determine the value of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\).

4 The roots of the equation \(2 x ^ { 3 } - 5 x + 7 = 0\) are \(\alpha , \beta\) and \(\gamma\).

1. Find \(\frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma }\).
2. Find an equation with integer coefficients whose roots are \(2 \alpha - 1,2 \beta - 1\) and \(2 \gamma - 1\).