Parametric polynomials with root conditions

5

1. Solve the equation \(z ^ { 2 } - 2 p \mathrm { i } z - q = 0\), where \(p\) and \(q\) are real constants.
   In an Argand diagram with origin \(O\), the roots of this equation are represented by the distinct points \(A\) and \(B\).
2. Given that \(A\) and \(B\) lie on the imaginary axis, find a relation between \(p\) and \(q\).
3. Given instead that triangle \(O A B\) is equilateral, express \(q\) in terms of \(p\).

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 \(z\).

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

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

1. Write down the other complex root. [1]
2. Explain why the equation must have a real root. [1]
3. Find the value of the real root and the values of \(A\) and \(B\). [9]

The cubic equation \(3z^3 + pz^2 + 17z + q = 0\), 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. [1 mark]
   2. Hence find the value of \(\alpha\beta\). [1 mark]
2. Find the value of the third root, \(\gamma\), of this equation. [3 marks]
3. Find the values of \(p\) and \(q\). [3 marks]

\(p(z) = z^4 + 3z^2 + az + b\), \(a \in \mathbb{R}\), \(b \in \mathbb{R}\) \(2 - 3i\) is a root of the equation \(p(z) = 0\)

1. Express \(p(z)\) as a product of quadratic factors with real coefficients. [5 marks]
2. Solve the equation \(p(z) = 0\). [1 mark]

In this question you must show detailed reasoning.

1. Explain why all cubic equations with real coefficients have at least one real root. [2]
2. Points representing the three roots of the equation \(z^3 + 9z^2 + 27z + 35 = 0\) are plotted on an Argand diagram. Find the exact area of the triangle which has these three points as its vertices. [7]