Complex roots with real coefficients

9 The quartic equation \(x ^ { 4 } + A x ^ { 3 } + B x ^ { 2 } + C x + D = 0\), where \(A , B , C\) and \(D\) are real numbers, has roots \(2 + \mathrm { j }\) and - 2 j .

1. Write down the other roots of the equation.
2. Find the values of \(A , B , C\) and \(D\).

9 The cubic equation \(x ^ { 3 } + A x ^ { 2 } + B x + 15 = 0\), where \(A\) and \(B\) are real numbers, has a root \(x = 1 + 2 \mathrm { j }\).

1. Write down the other complex root.
2. Explain why the equation must have a real root.
3. Find the value of the real root and the values of \(A\) and \(B\).

9 Two complex numbers, \(\alpha\) and \(\beta\), are given by \(\alpha = 2 - 2 \mathrm { j }\) and \(\beta = - 1 + \mathrm { j }\).
\(\alpha\) and \(\beta\) are both roots of a quartic equation \(x ^ { 4 } + A x ^ { 3 } + B x ^ { 2 } + C x + D = 0\), where \(A , B , C\) and \(D\) are real numbers.

1. Write down the other two roots.
2. Represent these four roots on an Argand diagram.
3. Find the values of \(A , B , C\) and \(D\).

6 One root of the cubic equation \(x ^ { 3 } + p x ^ { 2 } + 6 x + q = 0\), where \(p\) and \(q\) are real, is the complex number 5-i.

1. Find the real root of the cubic equation.
2. Find the values of \(p\) and \(q\).

9 One root of the quadratic equation \(x ^ { 2 } + a x + b = 0\), where \(a\) and \(b\) are real, is \(16 - 30 \mathrm { i }\).

1. Write down the other root of the quadratic equation.
2. Find the values of \(a\) and \(b\).
3. Use an algebraic method to solve the quartic equation \(y ^ { 4 } + a y ^ { 2 } + b = 0\).

8 The function \(\mathrm { f } ( z ) = z ^ { 4 } - z ^ { 3 } + a z ^ { 2 } + b z + c\) has real coefficients. The equation \(\mathrm { f } ( z ) = 0\) has roots \(\alpha , \beta\), \(\gamma\) and \(\delta\) where \(\alpha = 1\) and \(\beta = 1 + \mathrm { j }\).

1. Write down the other complex root and explain why the equation must have a second real root.
2. Write down the value of \(\alpha + \beta + \gamma + \delta\) and find the second real root.
3. Find the values of \(a , b\) and \(c\).
4. Write down \(\mathrm { f } ( - z )\) and the roots of \(\mathrm { f } ( - z ) = 0\).

3 You are given that \(z = 2 + \mathrm { j }\) is a root of the cubic equation \(2 z ^ { 3 } + p z ^ { 2 } + 22 z - 15 = 0\), where \(p\) is real. Find the other roots and the value of \(p\).

3 The cubic equation \(2 z ^ { 3 } - z ^ { 2 } + 4 z + k = 0\), where \(k\) is real, has a root \(z = 1 + 2 \mathrm { j }\).
Write down the other complex root. Hence find the real root and the value of \(k\).

8

1. Verify that \(1 + 3 \mathrm { j }\) is a root of the equation \(3 z ^ { 3 } - 2 z ^ { 2 } + 22 z + 40 = 0\), showing your working.
2. Explain why the equation must have exactly one real root.
3. Find the other roots of the equation.

3 You are given that \(z = 2 + 3 \mathrm { j }\) is a root of the quartic equation \(z ^ { 4 } - 5 z ^ { 3 } + 15 z ^ { 2 } - 5 z - 26 = 0\). Find the other roots.

7 The function \(\mathrm { f } ( z ) = 2 z ^ { 4 } - 9 z ^ { 3 } + A z ^ { 2 } + B z - 26\) has real coefficients. The equation \(\mathrm { f } ( z ) = 0\) has two real roots, \(\alpha\) and \(\beta\), where \(\alpha > \beta\), and two complex roots, \(\gamma\) and \(\delta\), where \(\gamma = 3 + 2 \mathrm { j }\).

1. Show that \(\alpha + \beta = - \frac { 3 } { 2 }\) and find the value of \(\alpha \beta\).
2. Hence find the two real roots \(\alpha\) and \(\beta\).
3. Find the values of \(A\) and \(B\).
4. Write down the roots of the equation \(\mathrm { f } \left( \frac { w } { \mathrm { j } } \right) = 0\).

3 In this question you must show detailed reasoning.
You are given that \(x = 2 + 5 \mathrm { i }\) is a root of the equation \(x ^ { 3 } - 2 x ^ { 2 } + 21 x + 58 = 0\).
Solve the equation.

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 \(\tan \theta = \sqrt { 2 }\).

2 In this question you must show detailed reasoning. The equation \(\mathrm { f } ( x ) = 0\), where \(\mathrm { f } ( x ) = x ^ { 4 } + 2 x ^ { 3 } + 2 x ^ { 2 } + 26 x + 169\), has a root \(x = 2 + 3 \mathrm { i }\).

1. Express \(\mathrm { f } ( x )\) as a product of two quadratic factors.
2. Hence write down all the roots of the equation \(\mathrm { f } ( x ) = 0\).

6. (a) Show that \(1 - 2 \mathrm { i }\) is a root of the cubic equation \(x ^ { 3 } + 5 x ^ { 2 } - 9 x + 35 = 0\).
(b) Find the other two roots of the equation.

3

1. Show that \(( 2 + \mathrm { i } ) ^ { 3 }\) can be expressed in the form \(2 + b \mathrm { i }\), where \(b\) is an integer.
2. It is given that \(2 + \mathrm { i }\) is a root of the equation $$z ^ { 3 } + p z + q = 0$$ where \(p\) and \(q\) are real numbers.
   1. Show that \(p = - 11\) and find the value of \(q\).
   2. Given that \(2 - \mathrm { i }\) is also a root of \(z ^ { 3 } + p z + q = 0\), find a quadratic factor of \(z ^ { 3 } + p z + q\) with real coefficients.
   3. Find the real root of the equation \(z ^ { 3 } + p z + q = 0\).
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3 The cubic equation $$2 z ^ { 3 } + p z ^ { 2 } + q z + 16 = 0$$ where \(p\) and \(q\) are real, has roots \(\alpha , \beta\) and \(\gamma\).
It is given that \(\alpha = 2 + 2 \sqrt { 3 } \mathrm { i }\).

1. 1. Write down another root, \(\beta\), of the equation.
   2. Find the third root, \(\gamma\).
   3. Find the values of \(p\) and \(q\).
2. 1. Express \(\alpha\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
   2. Show that $$( 2 + 2 \sqrt { 3 } \mathrm { i } ) ^ { n } = 4 ^ { n } \left( \cos \frac { n \pi } { 3 } + \mathrm { i } \sin \frac { n \pi } { 3 } \right)$$
   3. Show that $$\alpha ^ { n } + \beta ^ { n } + \gamma ^ { n } = 2 ^ { 2 n + 1 } \cos \frac { n \pi } { 3 } + \left( - \frac { 1 } { 2 } \right) ^ { n }$$ where \(n\) is an integer.

3

1. Show that \(( 1 + \mathrm { i } ) ^ { 3 } = 2 \mathrm { i } - 2\).
2. The cubic equation $$z ^ { 3 } - ( 5 + \mathrm { i } ) z ^ { 2 } + ( 9 + 4 \mathrm { i } ) z + k ( 1 + \mathrm { i } ) = 0$$ where \(k\) is a real constant, has roots \(\alpha , \beta\) and \(\gamma\).
   It is given that \(\alpha = 1 + \mathrm { i }\).
   1. Find the value of \(k\).
   2. Show that \(\beta + \gamma = 4\).
   3. 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 \(\frac { 1 } { \gamma }\).

2 The cubic equation \(3 z ^ { 3 } + p z ^ { 2 } + 17 z + q = 0\), where \(p\) and \(q\) are real, has a root \(\alpha = 1 + 2 \mathrm { i }\).

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

7 In the quartic equation \(2 x ^ { 4 } - 20 x ^ { 3 } + a x ^ { 2 } + b x + 250 = 0\), the coefficients \(a\) and \(b\) are real. One root of the equation is \(2 + \mathrm { i }\). Find the other roots.

4 You are given that \(z = 1 + 2 \mathrm { i }\) is a root of the equation \(z ^ { 3 } - 5 z ^ { 2 } + q z - 15 = 0\), where \(q \in \mathbb { R }\). Find

• the other roots,
• the value of \(q\).

4. The roots of the equation $$x ^ { 3 } - 4 x ^ { 2 } + 14 x - 20 = 0$$ are denoted by \(\alpha , \beta , \gamma\).

1. Show that $$\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = - 12$$ Explain why this result shows that exactly one of the roots of the above cubic equation is real.
2. Given that one of the roots is \(1 + 3 \mathrm { i }\), find the other two roots. Explain your method for each root.

7. $$f ( z ) = z ^ { 4 } - 6 z ^ { 3 } + p z ^ { 2 } + q z + r$$ where \(p , q\) and \(r\) are real constants.
The roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\) are \(\alpha , \beta , \gamma\) and \(\delta\) where \(\alpha = 3\) and \(\beta = 2 + \mathrm { i }\)
Given that \(\gamma\) is a complex root of \(\mathrm { f } ( \mathrm { z } ) = 0\)

1. 1. write down the root \(\gamma\),
   2. explain why \(\delta\) must be real.
2. Determine the value of \(\delta\).
3. Hence determine the values of \(p , q\) and \(r\).
4. Write down the roots of the equation \(\mathrm { f } ( - 2 \mathrm { z } ) = 0\)

1. \(\mathrm { f } ( \mathrm { z } ) = \mathrm { z } ^ { 3 } + a \mathrm { z } ^ { 2 } + b \mathrm { z } + 175 \quad\) where \(a\) and \(b\) are real constants

Given that \(- 3 + 4 \mathrm { i }\) is a root of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)

1. determine the value of \(a\) and the value of \(b\).
2. Show all the roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\) on a single Argand diagram.
3. Write down the roots of the equation \(\mathrm { f } ( \mathrm { z } + 2 ) = 0\)