Complex roots with real coefficients

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]

\(2 - 3i\) is one root of the equation $$z^3 + mz + 52 = 0$$ where \(m\) is real.

1. Find the other roots. [3 marks]
2. Determine the value of \(m\). [2 marks]

It is given that \(z = -\frac{3}{2} + i\frac{\sqrt{11}}{2}\) is a root of the equation $$z^4 - 3z^3 - 5z^2 + kz + 40 = 0$$ where \(k\) is a real number.

1. Find the other three roots. [5 marks]
2. Given that \(x \in \mathbb{R}\), solve $$x^4 - 3x^3 - 5x^2 + kx + 40 < 0$$ [1 mark]

Abel and Bonnie are trying to solve this mathematical problem: \(z = 2 - 3\mathrm{i}\) is a root of the equation \(2z^3 + mz^2 + pz + 91 = 0\) Find the value of \(m\) and the value of \(p\). Abel says he has solved the problem. Bonnie says there is not enough information to solve the problem.

1. Abel's solution begins as follows: Since \(z = 2 - 3\mathrm{i}\) is a root of the equation, \(z = 2 + 3\mathrm{i}\) is another root. State one extra piece of information about \(m\) and \(p\) which could be added to the problem to make the beginning of Abel's solution correct. [1 mark]
2. Prove that Bonnie is right. [4 marks]

Find a cubic equation with real coefficients, two of whose roots are \(2 - i\) and \(3\). [5]

You are given that \(z = 1 + 2i\) is a root of the equation \(z^3 - 5z^2 + qz - 15 = 0\), where \(q \in \mathbb{R}\). Find • the other roots, • the value of \(q\). [5]

The roots of the equation $$x^3 - 4x^2 + 14x - 20 = 0$$ 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. [3]
2. Given that one of the roots is \(1 + 3i\), find the other two roots. Explain your method for each root. [4]

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

1. Express f(x) as a product of two quadratic factors. [4]
2. Hence write down all the roots of the equation f(x) = 0. [1]

It is given that \(1 - 3i\) is one root of the quartic equation $$z^4 - 2z^3 + pz^2 + rz + 80 = 0$$ where \(p\) and \(r\) are real numbers.

1. Express \(z^4 - 2z^3 + pz^2 + rz + 80\) as the product of two quadratic factors with real coefficients. [4 marks]
2. Find the value of \(p\) and the value of \(r\). [2 marks]