4.02g Conjugate pairs: real coefficient polynomials

1. $$f ( z ) = 3 z ^ { 3 } + p z ^ { 2 } + 57 z + q$$ where \(p\) and \(q\) are real constants.
Given that \(3 - 2 \sqrt { 2 } \mathrm { i }\) is a root of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)

1. show all the roots of \(f ( z ) = 0\) on a single Argand diagram,
2. find the value of \(p\) and the value of \(q\).

1. $$\mathrm { f } ( \mathrm { z } ) = \mathrm { z } ^ { 3 } + a \mathrm { z } + 52 \quad \text { where } a \text { is a real constant }$$ Given that \(2 - 3 \mathrm { i }\) is a root of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)

1. write down the other complex root.
2. Hence
   1. solve completely \(\mathrm { f } ( \mathrm { z } ) = 0\)
   2. determine the value of \(a\)
3. Show all the roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\) on a single Argand diagram.

1. $$\mathrm { f } ( z ) = z ^ { 4 } - 6 z ^ { 3 } + a z ^ { 2 } + b z + 145$$ where \(a\) and \(b\) are real constants.
Given that \(2 + 5 \mathrm { i }\) is a root of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)

1. determine the other roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)
2. Show all the roots of \(\mathrm { f } ( \mathrm { z } ) = 0\) on a single Argand diagram.

3. $$\mathrm { f } ( z ) = z ^ { 4 } + a z ^ { 3 } + 6 z ^ { 2 } + b z + 65$$ where \(a\) and \(b\) are real constants.
Given that \(z = 3 + 2 \mathbf { i }\) is a root of the equation \(\mathrm { f } ( z ) = 0\), show the roots of \(\mathrm { f } ( z ) = 0\) on a single Argand diagram.

5 Throughout this question the use of a calculator is not permitted. It is given that the complex number \(- 1 + ( \sqrt { } 3 ) \mathrm { i }\) is a root of the equation $$k x ^ { 3 } + 5 x ^ { 2 } + 10 x + 4 = 0$$ where \(k\) is a real constant.

1. Write down another root of the equation.
2. Find the value of \(k\) and the third root of the equation.

9 In this question you must show detailed reasoning. You are given that \(a\) is a real root of the equation \(x ^ { 4 } + x ^ { 3 } + 3 x ^ { 2 } - 5 x = 0\).
You are also given that \(a + 2 + 3 \mathrm { i }\) is one root of the equation \(z ^ { 4 } - 2 ( 1 + a ) z ^ { 3 } + ( 21 a - 10 ) z ^ { 2 } + ( 86 - 80 a ) z + ( 285 a - 195 ) = 0\). Determine all possible values of \(z\).

10 In this question you must show detailed reasoning.

1. You are given that \(- 1 + \mathrm { i }\) is a root of the equation \(z ^ { 3 } = a + b \mathrm { i }\), where \(a\) and \(b\) are real numbers. Find \(a\) and \(b\).
2. Find all the roots of the equation in part (a), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r\) and \(\theta\) are exact.
3. Chris says "the complex roots of a polynomial equation come in complex conjugate pairs". Explain why this does not apply to the polynomial equation in part (a).

8 In this question you must show detailed reasoning. You are given that i is a root of the equation \(z ^ { 4 } - 2 z ^ { 3 } + 3 z ^ { 2 } + a z + b = 0\), where \(a\) and \(b\) are real constants.

1. Show that \(a = - 2\) and \(b = 2\).
2. Find the other roots of this equation.

7 In this question you must show detailed reasoning.

1. Find the square roots of the number \(528 + 46 \mathrm { i }\) giving your answers in the form \(a + b \mathrm { i }\).
2. \(\quad 3 + 2 \mathrm { i }\) is a root of the equation \(x ^ { 3 } - a x + 78 = 0\), where \(a\) is a real number. Find the value of \(a\).

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 \(r\).

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 \(q\).

8 Stephen is correctly told that \(( 1 + \mathrm { i } )\) and - 1 are two roots of the polynomial equation $$z ^ { 3 } - 2 \mathrm { i } z ^ { 2 } + p z + q = 0$$ where \(p\) and \(q\) are complex numbers.
8

1. Stephen states that ( \(1 - \mathrm { i }\) ) must also be a root of the equation because roots of polynomial equations occur in conjugate pairs. Explain why Stephen's reasoning is wrong. 8
2. \(\quad\) Find \(p\) and \(q\)

4

1. Express \(z ^ { 4 } - 2 z ^ { 3 } + p z ^ { 2 } + r z + 80\) as the product of two quadratic factors with real coefficients.
   [4 marks]
   4 It is given that \(1 - 3 \mathrm { i }\) is one root of the quartic equation
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   4
2. Find the value of \(p\) and the value of \(r\).

3 In this question you must show detailed reasoning.
The equation \(x ^ { 4 } - 7 x ^ { 3 } - 2 x ^ { 2 } + 218 x - 1428 = 0\) has a root \(3 - 5 i\).
Find the other three roots of this equation.

2 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.

6 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 \(\alpha = 1 + i \sqrt { 2 }\).

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 roots of the equation \(z ^ { 2 } - 6 z + 10 = 0\) are \(z _ { 1 }\) and \(z _ { 2 }\), where \(z _ { 1 } = 3 + \mathrm { i }\).

1. Write down the value of \(z _ { 2 }\).
2. Show \(z _ { 1 }\) and \(z _ { 2 }\) on an Argand diagram.
3. Show that \(z _ { 1 } ^ { 2 } = 8 + 6 \mathrm { i }\).

5 A root of the equation \(z ^ { 2 } + p z + q = 0\) is \(3 + \mathrm { i }\), where \(p\) and \(q\) are real. Write down the other root of the equation and hence calculate the values of \(p\) and \(q\).

9

1. Show that \(z = ( 1 + \mathrm { i } )\) is a root of the cubic equation \(3 z ^ { 3 } - 8 z ^ { 2 } + 10 z - 4 = 0\).
2. Show that the equation \(3 z ^ { 3 } - 8 z ^ { 2 } + 10 z - 4 = 0\) has a quadratic factor with real coefficients and hence solve this equation completely.

Throughout this question the use of a calculator is not permitted. The complex number \(2 - \mathrm{i}\) is denoted by \(u\).

1. It is given that \(u\) is a root of the equation \(x^3 + ax^2 - 3x + b = 0\), where the constants \(a\) and \(b\) are real. Find the values of \(a\) and \(b\). [4]
2. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying both the inequalities \(|z - u| < 1\) and \(|z| < |z + \mathrm{i}|\). [4]

The equation $$x^4 + Ax^3 + Bx^2 + Cx + 225 = 0$$ where \(A\), \(B\) and \(C\) are real constants, has

• a complex root \(4 + 3\text{i}\)
• a repeated positive real root

1. Write down the other complex root of this equation. [1]
2. Hence determine a quadratic factor of \(x^4 + Ax^3 + Bx^2 + Cx + 225\) [2]
3. Deduce the real root of the equation. [2]
4. Hence determine the value of each of the constants \(A\), \(B\) and \(C\) [3]