Find conjugate roots from polynomial

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1. It is given that \(- 1 + ( \sqrt { } 5 ) \mathrm { i }\) is a root of the equation \(z ^ { 3 } + 2 z + a = 0\), where \(a\) is real. Showing your working, find the value of \(a\), and write down the other complex root of this equation.
2. The complex number \(w\) has modulus 1 and argument \(2 \theta\) radians. Show that \(\frac { w - 1 } { w + 1 } = \mathrm { i } \tan \theta\).

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1. Verify that \(- 1 + \sqrt { 2 } \mathrm { i }\) is a root of the equation \(z ^ { 4 } + 3 z ^ { 2 } + 2 z + 12 = 0\).
2. Find the other roots of this equation.
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1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

Given that \(x = 2 + 3 \mathrm { i }\) is a root of the equation $$2 x ^ { 4 } - 8 x ^ { 3 } + 29 x ^ { 2 } - 12 x + 39 = 0$$

1. write down another complex root of this equation.
2. Use algebra to determine the other 2 roots of the equation.
3. Show all 4 roots on a single Argand diagram.

8. $$\mathrm { f } ( z ) = z ^ { 4 } + 6 z ^ { 3 } + 76 z ^ { 2 } + a z + b$$ where \(a\) and \(b\) are real constants.
Given that \(- 3 + 8 \mathrm { i }\) is a complex root of the equation \(\mathrm { f } ( z ) = 0\)

1. write down another complex root of this equation.
2. Hence, or otherwise, find the other roots of the equation \(\mathrm { f } ( z ) = 0\)
3. Show on a single Argand diagram all four roots of the equation \(\mathrm { f } ( z ) = 0\)

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4 （a）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 (b) Find the value of \(p\) and the value of \(r\).