Given one complex root of cubic or quartic, find all roots

5. The roots of the equation $$z ^ { 3 } - 8 z ^ { 2 } + 22 z - 20 = 0$$ are \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\).

1. Given that \(z _ { 1 } = 3 + \mathrm { i }\), find \(z _ { 2 }\) and \(z _ { 3 }\).
2. Show, on a single Argand diagram, the points representing \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\).

8 You are given that the complex number \(\alpha = 1 + \mathrm { j }\) satisfies the equation \(z ^ { 3 } + 3 z ^ { 2 } + p z + q = 0\), where \(p\) and \(q\) are real constants.

1. Find \(\alpha ^ { 2 }\) and \(\alpha ^ { 3 }\) in the form \(a + b \mathrm { j }\). Hence show that \(p = - 8\) and \(q = 10\).
2. Find the other two roots of the equation.
3. Represent the three roots on an Argand diagram.

3. Given that \(5 - \mathrm { i }\) is a root of the equation \(x ^ { 4 } - 10 x ^ { 3 } + 10 x ^ { 2 } + 160 x - 416 = 0\),

1. write down another root of the equation,
2. find the remaining roots.

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.

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.

4 In this question you must show detailed reasoning. You are given that \(\mathrm { f } ( \mathrm { z } ) = 4 \mathrm { z } ^ { 4 } - 12 \mathrm { z } ^ { 3 } + 41 \mathrm { z } ^ { 2 } - 128 \mathrm { z } + 185\) and that \(2 + \mathrm { i }\) is a root of the equation \(f ( z ) = 0\).

1. Express \(\mathrm { f } ( \mathrm { z } )\) as the product of two quadratic factors with integer coefficients.
2. Solve \(f ( z ) = 0\). Two loci on an Argand diagram are defined by \(C _ { 1 } = \left\{ z : | z | = r _ { 1 } \right\}\) and \(C _ { 2 } = \left\{ z : | z | = r _ { 2 } \right\}\) where \(r _ { 1 } > r _ { 2 }\). You are given that two of the points representing the roots of \(\mathrm { f } ( \mathrm { z } ) = 0\) are on \(C _ { 1 }\) and two are on \(C _ { 2 } . R\) is the region on the Argand diagram between \(C _ { 1 }\) and \(C _ { 2 }\).
3. Find the exact area of \(R\).
4. \(\omega\) is the sum of all the roots of \(\mathrm { f } ( \mathrm { z } ) = 0\). Determine whether or not the point on the Argand diagram which represents \(\omega\) lies in \(R\).

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 \(\mathrm { f } ( z ) = 4 z ^ { 4 } - 12 z ^ { 3 } + 41 z ^ { 2 } - 128 z + 185\) and that \(2 + \mathrm { i }\) is a root of the equation \(\mathrm { f } ( z ) = 0\).

1. Express \(\mathrm { f } ( z )\) as the product of two quadratic factors with integer coefficients.
2. Solve \(\mathrm { f } ( z ) = 0\). Two loci on an Argand diagram are defined by \(C _ { 1 } = \left\{ z : | z | = r _ { 1 } \right\}\) and \(C _ { 2 } = \left\{ z : | z | = r _ { 2 } \right\}\) where \(r _ { 1 } > r _ { 2 }\). You are given that two of the points representing the roots of \(\mathrm { f } ( z ) = 0\) are on \(C _ { 1 }\) and two are on \(C _ { 2 } \cdot R\) is the region on the Argand diagram between \(C _ { 1 }\) and \(C _ { 2 }\).
3. Find the exact area of \(R\).
4. \(\omega\) is the sum of all the roots of \(\mathrm { f } ( z ) = 0\). Determine whether or not the point on the Argand diagram which represents \(\omega\) lies in \(R\).

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.

Given that \(1 - i\) is a root of the equation \(z^3 - 3z^2 + 4z - 2 = 0\), find the other two roots. Tick \((\checkmark)\) one box. [1 mark] \(-1 + i\) and \(-1\) \(1 + i\) and \(1\) \(-1 + i\) and \(1\) \(1 + i\) and \(-1\)

In this question you must show detailed reasoning. You are given that \(f(z) = 4z^4 - 12z^3 + 41z^2 - 128z + 185\) and that \(2 + \mathrm{i}\) is a root of the equation \(f(z) = 0\).

1. Express \(f(z)\) as the product of two quadratic factors with integer coefficients. [5]
2. Solve \(f(z) = 0\). [3] Two loci on an Argand diagram are defined by \(C_1 = \{z:|z| = r_1\}\) and \(C_2 = \{z:|z| = r_2\}\) where \(r_1 > r_2\). You are given that two of the points representing the roots of \(f(z) = 0\) are on \(C_1\) and two are on \(C_2\). \(R\) is the region on the Argand diagram between \(C_1\) and \(C_2\).
3. Find the exact area of \(R\). [4]
4. \(\omega\) is the sum of all the roots of \(f(z) = 0\). Determine whether or not the point on the Argand diagram which represents \(\omega\) lies in \(R\). [2]