4.05a Roots and coefficients: symmetric functions

8 In this question you must show detailed reasoning. The equation \(\mathrm { x } ^ { 3 } + \mathrm { kt } ^ { 2 } + 15 \mathrm { x } - 25 = 0\) has roots \(\alpha , \beta\) and \(\frac { \alpha } { \beta }\). Given that \(\alpha > 0\), find, in any order,

• the roots of the equation,
• the value of \(k\).

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 \(\gamma \alpha\).

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

1. Evaluate, in exact form, the roots of the equation.
2. Find \(k\).

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 In this question you must show detailed reasoning. The equation \(z ^ { 3 } + 2 z ^ { 2 } + k z + 3 = 0\), where \(k\) is a constant, has roots \(\alpha , \frac { 1 } { \alpha }\) and \(\beta\).
Determine the roots in exact form.

2 In this question you must show detailed reasoning.
The quadratic equation \(3 x ^ { 2 } - 7 x + 5 = 0\) has roots \(\alpha\) and \(\beta\).

1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
2. Hence find the values of the following expressions.
   1. \(\frac { 1 } { \alpha } + \frac { 1 } { \beta }\)
   2. \(\alpha ^ { 2 } + \beta ^ { 2 }\) \(3 \quad l _ { 1 }\) and \(l _ { 2 }\) are two intersecting straight lines with the following equations. $$\begin{aligned} & l _ { 1 } : \mathbf { r } = \left( \begin{array} { c } 3 \\ 3 \\ - 5 \end{array} \right) \end{aligned}$$

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

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

8

1. 1. It is given that \(\alpha\) and \(\beta\) are the roots of the equation $$x ^ { 2 } - 2 x + 4 = 0$$ Without solving this equation, show that \(\alpha ^ { 3 }\) and \(\beta ^ { 3 }\) are the roots of the equation $$x ^ { 2 } + 16 x + 64 = 0$$ (6 marks)
   2. State, giving a reason, whether the roots of the equation $$x ^ { 2 } + 16 x + 64 = 0$$ are real and equal, real and distinct, or non-real.
2. Solve the equation $$x ^ { 2 } - 2 x + 4 = 0$$
3. Use your answers to parts (a) and (b) to show that $$( 1 + \mathrm { i } \sqrt { 3 } ) ^ { 3 } = ( 1 - \mathrm { i } \sqrt { 3 } ) ^ { 3 }$$

1 The quadratic equation $$3 x ^ { 2 } - 6 x + 1 = 0$$ has roots \(\alpha\) and \(\beta\).

1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
2. Show that \(\alpha ^ { 3 } + \beta ^ { 3 } = 6\).
3. Find a quadratic equation, with integer coefficients, which has roots \(\frac { \alpha ^ { 2 } } { \beta }\) and \(\frac { \beta ^ { 2 } } { \alpha }\).

6 The equation $$x ^ { 2 } - 4 x + 13 = 0$$ has roots \(\alpha\) and \(\beta\).

1. 1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
   2. Deduce that \(\alpha ^ { 2 } + \beta ^ { 2 } = - 10\).
   3. Explain why the statement \(\alpha ^ { 2 } + \beta ^ { 2 } = - 10\) implies that \(\alpha\) and \(\beta\) cannot both be real.
2. Find in the form \(p + \mathrm { i } q\) the values of:
   1. \(( \alpha + \mathrm { i } ) + ( \beta + \mathrm { i } )\);
   2. \(( \alpha + \mathrm { i } ) ( \beta + \mathrm { i } )\).
3. Hence find a quadratic equation with roots \(( \alpha + \mathrm { i } )\) and \(( \beta + \mathrm { i } )\).

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

1. Write down the numerical values of \(\alpha + \beta\) and \(\alpha \beta\).
2. 1. Expand \(( \alpha + \beta ) ^ { 3 }\).
   2. Show that \(\alpha ^ { 3 } + \beta ^ { 3 } = 4\).
3. Find a quadratic equation with roots \(\alpha ^ { 3 }\) and \(\beta ^ { 3 }\), giving your answer in the form \(p x ^ { 2 } + q x + r = 0\), where \(p , q\) and \(r\) are integers.

2 The cubic equation $$x ^ { 3 } + p x ^ { 2 } + q x + r = 0$$ where \(p , q\) and \(r\) are real, has roots \(\alpha , \beta\) and \(\gamma\).

1. Given that $$\alpha + \beta + \gamma = 4 \quad \text { and } \quad \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 20$$ find the values of \(p\) and \(q\).
2. Given further that one root is \(3 + \mathrm { i }\), find the value of \(r\).

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 \(\gamma ^ { 2 }\).

4 It is given that \(\alpha , \beta\) and \(\gamma\) satisfy the equations $$\begin{aligned} & \alpha + \beta + \gamma = 1 \\ & \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = - 5 \\ & \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = - 23 \end{aligned}$$

1. Show that \(\alpha \beta + \beta \gamma + \gamma \alpha = 3\).
2. Use the identity $$( \alpha + \beta + \gamma ) \left( \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } - \alpha \beta - \beta \gamma - \gamma \alpha \right) = \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } - 3 \alpha \beta \gamma$$ to find the value of \(\alpha \beta \gamma\).
3. Write down a cubic equation, with integer coefficients, whose roots are \(\alpha , \beta\) and \(\gamma\).
4. Explain why this cubic equation has two non-real roots.
5. Given that \(\alpha\) is real, find the values of \(\alpha , \beta\) and \(\gamma\).

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 \(\gamma\).

2 The cubic equation $$z ^ { 3 } + p z ^ { 2 } + 6 z + q = 0$$ has roots \(\alpha , \beta\) and \(\gamma\).

1. Write down the value of \(\alpha \beta + \beta \gamma + \gamma \alpha\).
2. Given that \(p\) and \(q\) are real and that \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = - 12\) :
   1. explain why the cubic equation has two non-real roots and one real root;
   2. find the value of \(p\).
3. One root of the cubic equation is \(- 1 + 3 \mathrm { i }\). Find:
   1. the other two roots;
   2. the value of \(q\).

3 The cubic equation $$z ^ { 3 } + p z ^ { 2 } + 25 z + q = 0$$ where \(p\) and \(q\) are real, has a root \(\alpha = 2 - 3 \mathrm { i }\).

1. Write down another non-real root, \(\beta\), of this equation.
2. Find:
   1. the value of \(\alpha \beta\);
   2. the third root, \(\gamma\), of the equation;
   3. the values of \(p\) and \(q\).

7 The cubic equation \(27 z ^ { 3 } + k z ^ { 2 } + 4 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).

1. Write down the values of \(\alpha \beta + \beta \gamma + \gamma \alpha\) and \(\alpha \beta \gamma\).
2. 1. In the case where \(\beta = \gamma\), find the roots of the equation.
   2. Find the value of \(k\) in this case.
3. 1. In the case where \(\alpha = 1 - \mathrm { i }\), find \(\alpha ^ { 2 }\) and \(\alpha ^ { 3 }\).
   2. Hence find the value of \(k\) in this case.
4. In the case where \(k = - 12\), find a cubic equation with integer coefficients which has roots \(\frac { 1 } { \alpha } + 1 , \frac { 1 } { \beta } + 1\) and \(\frac { 1 } { \gamma } + 1\).
   [0pt] [5 marks]

8 The complex number \(\omega\) is given by \(\omega = \cos \frac { 2 \pi } { 5 } + \mathrm { i } \sin \frac { 2 \pi } { 5 }\).

1. 1. Verify that \(\omega\) is a root of the equation \(z ^ { 5 } = 1\).
   2. Write down the three other non-real roots of \(z ^ { 5 } = 1\), in terms of \(\omega\).
2. 1. Show that \(1 + \omega + \omega ^ { 2 } + \omega ^ { 3 } + \omega ^ { 4 } = 0\).
   2. Hence show that \(\left( \omega + \frac { 1 } { \omega } \right) ^ { 2 } + \left( \omega + \frac { 1 } { \omega } \right) - 1 = 0\).
3. Hence show that \(\cos \frac { 2 \pi } { 5 } = \frac { \sqrt { 5 } - 1 } { 4 }\).

13

1. Given that $$P ( x ) = 125 x ^ { 3 } + 150 x ^ { 2 } + 55 x + 6$$ use the factor theorem to prove that ( \(5 x + 1\) ) is a factor of \(\mathrm { P } ( x )\).
   [0pt] [2 marks]
   13
2. Factorise \(\mathrm { P } ( x )\) completely.
   13
3. Hence, prove that \(250 n ^ { 3 } + 300 n ^ { 2 } + 110 n + 12\) is a multiple of 12 when \(n\) is a positive whole number.

2 The quadratic equation \(x ^ { 2 } + p x + q = 0\) has roots \(\alpha\) and \(\beta\) Which of the following is equal to \(\alpha \beta\) ?
Circle your answer.
[0pt] [1 mark] \(p - p - q - q\)

11 The equation \(x ^ { 3 } - 8 x ^ { 2 } + c x + d = 0\) where \(c\) and \(d\) are real numbers, has roots \(\alpha , \beta , \gamma\).
When plotted on an Argand diagram, the triangle with vertices at \(\alpha , \beta , \gamma\) has an area of 8 . Given \(\alpha = 2\), find the values of \(c\) and \(d\). Fully justify your solution.
[0pt] [5 marks]

3 The quadratic equation \(a x ^ { 2 } + b x + c = 0 ( a , b , c \in \mathbb { R } )\) has real roots \(\alpha\) and \(\beta\). One of the four statements below is incorrect. Which statement is incorrect? Tick ( \(\checkmark\) ) one box. \(c = 0 \Rightarrow \alpha = 0\) or \(\beta = 0\) □ \(c = a \Rightarrow \alpha\) is the reciprocal of \(\beta\) □ \(b < 0\) and \(c < 0 \Rightarrow \alpha > 0\) and \(\beta > 0\) □ \(b = 0 \Rightarrow \alpha = - \beta\) □

8 The three roots of the equation $$4 x ^ { 3 } - 12 x ^ { 2 } - 13 x + k = 0$$ where \(k\) is a constant, form an arithmetic sequence. Find the roots of the equation.

3 The roots of the equation \(x ^ { 2 } - p x - 6 = 0\) are \(\alpha\) and \(\beta\) Find \(\alpha ^ { 2 } + \beta ^ { 2 }\) in terms of \(p\) Circle your answer. \(p ^ { 2 } - 6\) \(p ^ { 2 } + 6\) \(p ^ { 2 } - 12\) \(p ^ { 2 } + 12\)