4.05a Roots and coefficients: symmetric functions

6 Use de Moivre's theorem to show that $$\cos 4 \theta = 8 \cos ^ { 4 } \theta - 8 \cos ^ { 2 } \theta + 1$$ Without using a calculator, verify that \(\cos 4 \theta = - \cos 3 \theta\) for each of the values \(\theta = \frac { 1 } { 7 } \pi , \frac { 3 } { 7 } \pi , \frac { 5 } { 7 } \pi , \pi\). Using the result \(\cos 3 \theta = 4 \cos ^ { 3 } \theta - 3 \cos \theta\), show that the roots of the equation $$8 c ^ { 4 } + 4 c ^ { 3 } - 8 c ^ { 2 } - 3 c + 1 = 0$$ are \(\cos \frac { 1 } { 7 } \pi , \cos \frac { 3 } { 7 } \pi , \cos \frac { 5 } { 7 } \pi , - 1\). Deduce that \(\cos \frac { 1 } { 7 } \pi + \cos \frac { 3 } { 7 } \pi + \cos \frac { 5 } { 7 } \pi = \frac { 1 } { 2 }\).

The roots of the equation \(x ^ { 4 } - 3 x ^ { 2 } + 5 x - 2 = 0\) are \(\alpha , \beta , \gamma , \delta\), and \(\alpha ^ { n } + \beta ^ { n } + \gamma ^ { n } + \delta ^ { n }\) is denoted by \(S _ { n }\). Show that $$S _ { n + 4 } - 3 S _ { n + 2 } + 5 S _ { n + 1 } - 2 S _ { n } = 0$$ Find the values of

1. \(S _ { 2 }\) and \(S _ { 4 }\),
2. \(S _ { 3 }\) and \(S _ { 5 }\). Hence find the value of $$\alpha ^ { 2 } \left( \beta ^ { 3 } + \gamma ^ { 3 } + \delta ^ { 3 } \right) + \beta ^ { 2 } \left( \gamma ^ { 3 } + \delta ^ { 3 } + \alpha ^ { 3 } \right) + \gamma ^ { 2 } \left( \delta ^ { 3 } + \alpha ^ { 3 } + \beta ^ { 3 } \right) + \delta ^ { 2 } \left( \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } \right) .$$

5 The equation $$8 x ^ { 3 } + 36 x ^ { 2 } + k x - 21 = 0$$ where \(k\) is a constant, has roots \(a - d , a , a + d\). Find the numerical values of the roots and determine the value of \(k\).

The roots of the quartic equation \(x ^ { 4 } + 4 x ^ { 3 } + 2 x ^ { 2 } - 4 x + 1 = 0\) are \(\alpha , \beta , \gamma\) and \(\delta\). Find the values of

1. \(\alpha + \beta + \gamma + \delta\),
2. \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 }\),
3. \(\frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma } + \frac { 1 } { \delta }\),
4. \(\frac { \alpha } { \beta \gamma \delta } + \frac { \beta } { \alpha \gamma \delta } + \frac { \gamma } { \alpha \beta \delta } + \frac { \delta } { \alpha \beta \gamma }\). Using the substitution \(y = x + 1\), find a quartic equation in \(y\). Solve this quartic equation and hence find the roots of the equation \(x ^ { 4 } + 4 x ^ { 3 } + 2 x ^ { 2 } - 4 x + 1 = 0\).

2 Find the cubic equation with roots \(\alpha , \beta\) and \(\gamma\) such that $$\begin{aligned} \alpha + \beta + \gamma & = 3 \\ \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } & = 1 \\ \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } & = - 30 \end{aligned}$$ giving your answer in the form \(x ^ { 3 } + p x ^ { 2 } + q x + r = 0\), where \(p , q\) and \(r\) are integers to be found.

4 The cubic equation \(2 x ^ { 3 } - 3 x ^ { 2 } + 4 x - 10 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).

1. Find the value of \(( \alpha + 1 ) ( \beta + 1 ) ( \gamma + 1 )\).
2. Find the value of \(( \beta + \gamma ) ( \gamma + \alpha ) ( \alpha + \beta )\).

10

1. Use de Moivre's theorem to show that $$\sin 5 \theta = 5 \sin \theta - 20 \sin ^ { 3 } \theta + 16 \sin ^ { 5 } \theta$$
2. Hence explain why the roots of the equation \(16 x ^ { 4 } - 20 x ^ { 2 } + 5 = 0\) are \(x = \pm \sin \frac { 1 } { 5 } \pi\) and \(x = \pm \sin \frac { 2 } { 5 } \pi\).
3. Without using a calculator, find the exact values of $$\sin \frac { 1 } { 5 } \pi \sin \frac { 2 } { 5 } \pi \sin \frac { 3 } { 5 } \pi \sin \frac { 4 } { 5 } \pi \quad \text { and } \quad \sin ^ { 2 } \left( \frac { 1 } { 5 } \pi \right) + \sin ^ { 2 } \left( \frac { 2 } { 5 } \pi \right) .$$

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

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

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]

1 The quartic equation \(x ^ { 4 } - p x ^ { 2 } + q x - r = 0\), where \(p , q\) and \(r\) are real constants, has two pairs of equal roots. Show that \(p ^ { 2 } + 4 r = 0\) and state the value of \(q\).

4 The roots of the equation $$x ^ { 3 } - 8 x ^ { 2 } + 5 = 0$$ are \(\alpha , \beta , \gamma\). Show that $$\alpha ^ { 2 } = \frac { 5 } { \beta + \gamma } .$$ It is given that the roots are all real. Without reference to a graph, show that one of the roots is negative and the other two roots are positive.

1 The equation \(x ^ { 3 } + p x + q = 0\) has a repeated root. Prove that \(4 p ^ { 3 } + 27 q ^ { 2 } = 0\).

Let \(\omega = \cos \frac { 1 } { 5 } \pi + \mathrm { i } \sin \frac { 1 } { 5 } \pi\). Show that \(\omega ^ { 5 } + 1 = 0\) and deduce that $$\omega ^ { 4 } - \omega ^ { 3 } + \omega ^ { 2 } - \omega = - 1$$ Show further that $$\omega - \omega ^ { 4 } = 2 \cos \frac { 1 } { 5 } \pi \quad \text { and } \quad \omega ^ { 3 } - \omega ^ { 2 } = 2 \cos \frac { 3 } { 5 } \pi$$ Hence find the values of $$\cos \frac { 1 } { 5 } \pi + \cos \frac { 3 } { 5 } \pi \quad \text { and } \quad \cos \frac { 1 } { 5 } \pi \cos \frac { 3 } { 5 } \pi$$ Find a quadratic equation having roots \(\cos \frac { 1 } { 5 } \pi\) and \(\cos \frac { 3 } { 5 } \pi\) and deduce the exact value of \(\cos \frac { 1 } { 5 } \pi\).

7 A cubic equation has roots \(\alpha , \beta\) and \(\gamma\) such that $$\begin{aligned} \alpha + \beta + \gamma & = 4 \\ \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } & = 14 \\ \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } & = 34 \end{aligned}$$ Find the value of \(\alpha \beta + \beta \gamma + \gamma \alpha\). Show that the cubic equation is $$x ^ { 3 } - 4 x ^ { 2 } + x + 6 = 0$$ and solve this equation.

2 The cubic equation \(x ^ { 3 } - p x - q = 0\), where \(p\) and \(q\) are constants, has roots \(\alpha , \beta , \gamma\). Show that

1. \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 2 p\),
2. \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = 3 q\),
3. \(6 \left( \alpha ^ { 5 } + \beta ^ { 5 } + \gamma ^ { 5 } \right) = 5 \left( \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } \right) \left( \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } \right)\).

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 \(r\).

4 The roots of the equation $$5 x ^ { 3 } + 2 x ^ { 2 } - 3 x + p = 0$$ are \(\alpha , \beta\) and \(\gamma\) Given that \(p\) is a constant, state the value of \(\alpha \beta + \beta \gamma + \gamma \alpha\) Circle your answer. \(- \frac { 3 } { 5 }\) \(- \frac { 2 } { 5 }\) \(\frac { 2 } { 5 }\) \(\frac { 3 } { 5 }\)

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

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

1 The points \(A , B\) and \(C\) have position vectors \(\mathbf { a } = \left( \begin{array} { l } 1 \\ 1 \\ 3 \end{array} \right) , \mathbf { b } = \left( \begin{array} { r } 2 \\ 3 \\ - 1 \end{array} \right)\) and \(\mathbf { c } = \left( \begin{array} { r } - 5 \\ 1 \\ 2 \end{array} \right)\) respectively, relative to the origin \(O\).

1. Calculate, in its simplest exact form, the area of triangle \(O A B\).
2. Show that \(\mathbf { a } \times ( \mathbf { b } \times \mathbf { c } ) + \mathbf { b } \times ( \mathbf { c } \times \mathbf { a } ) + \mathbf { c } \times ( \mathbf { a } \times \mathbf { b } ) = \mathbf { 0 }\).

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

1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
2. Hence find a quadratic equation with roots \(\alpha + \frac { 1 } { \beta }\) and \(\beta + \frac { 1 } { \alpha }\).

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

9 You are given that the cubic equation \(2 x ^ { 3 } + p x ^ { 2 } + q x - 3 = 0\), where \(p\) and \(q\) are real numbers, has a complex root \(\alpha = 1 + \mathrm { i } \sqrt { 2 }\).

1. Write down a second complex root, \(\beta\).
2. Determine the third root, \(\gamma\).
3. Find the value of \(p\) and the value of \(q\).
4. Show that if \(n\) is an integer then \(\alpha ^ { n } + \beta ^ { n } + \gamma ^ { n } = 2 \times 3 ^ { \frac { 1 } { 2 } n } \times \cos n \theta + \frac { 1 } { 2 ^ { n } }\) where \(\tan \theta = \sqrt { 2 }\).

6 The equation \(x ^ { 3 } + 2 x ^ { 2 } + x + 3 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
The equation \(x ^ { 3 } + p x ^ { 2 } + q x + r = 0\) has roots \(\alpha \beta , \beta \gamma\) and \(\gamma \alpha\).
Find the values of \(p , q\) and \(r\).

10 The Argand diagram below shows the origin \(O\) and pentagon \(A B C D E\), where \(A , B , C , D\) and \(E\) are the points that represent the complex numbers \(a , b , c , d\) and \(e\), and where \(a\) is a positive real number. You are given that these five complex numbers are the roots of the equation \(z ^ { 5 } - a ^ { 5 } = 0\). \includegraphics[max width=\textwidth, alt={}, center]{94ecfc6e-df52-45a0-8f7b-f33fda391b15-4_903_883_477_502}

1. Justify each of the following statements.
   1. \(A , B , C , D\) and \(E\) lie on a circle with centre \(O\).
   2. \(A B C D E\) is a regular pentagon.
   3. \(b \times \mathrm { e } ^ { \frac { 2 \mathrm { i } \pi } { 5 } } = c\)
   4. \(b ^ { * } = e\)
   5. \(a + b + c + d + e = 0\)
   6. The midpoints of sides \(A B , B C , C D , D E\) and \(E A\) represent the complex numbers \(p , q , r , s\) and \(t\). Determine a polynomial equation, with real coefficients, that has roots \(p , q , r , s\) and \(t\).