4.02g Conjugate pairs: real coefficient polynomials

Given that \(2 + i\) is a root of the equation $$z^2 + bz + c = 0, \text{ where } b \text{ and } c \text{ are real constants,}$$

1. write down the other root of the equation,
2. find the value of \(b\) and the value of \(c\). [5]

Given that \(-2\) is a root of the equation \(z^3 + 6z + 20 = 0\),

1. Find the other two roots of the equation, [3]
2. show, on a single Argand diagram, the three points representing the roots of the equation, [1]
3. prove that these three points are the vertices of a right-angled triangle. [2]

$$z = \sqrt{3} - i.$$ \(z^*\) is the complex conjugate of \(z\).

1. Show that \(\frac{z}{z^*} = \frac{1}{2} - \frac{\sqrt{3}}{2} i\). [3]
2. Find the value of \(\left| \frac{z}{z^*} \right|\). [2]
3. Verify, for \(z = \sqrt{3} - i\), that \(\arg \frac{z}{z^*} = \arg z - \arg z^*\). [4]
4. Display \(z\), \(z^*\) and \(\frac{z}{z^*}\) on a single Argand diagram. [2]
5. Find a quadratic equation with roots \(z\) and \(z^*\) in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\) and \(c\) are real constants to be found. [2]

Given that \(x = -\frac{1}{2}\) is the real solution of the equation $$2x^3 - 11x^2 + 14x + 10 = 0,$$ find the two complex solutions of this equation. [6]

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]

\(2 - 3i\) 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]

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 \(-1\)

Given that \(z = 1 - 3\mathrm{i}\) is one root of the equation \(z^2 + pz + r = 0\), where \(p\) and \(r\) are real, find the value of \(r\). Circle your answer. [1 mark] \(-8\) \quad \(-2\) \quad \(6\) \quad \(10\)

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]

\(p(z) = z^4 + 3z^2 + az + b\), \(a \in \mathbb{R}\), \(b \in \mathbb{R}\) \(2 - 3i\) 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]

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]

The function f is a quartic function with real coefficients. The complex number \(5i\) is a root of the equation \(f(x) = 0\) Which one of the following must be a factor of \(f(x)\)? Circle your answer. [1 mark] \((x^2 - 25)\) \quad\quad \((x^2 - 5)\) \quad\quad \((x^2 + 5)\) \quad\quad \((x^2 + 25)\)

Latifa and Sam are studying polynomial equations of degree greater than 2, with real coefficients and no repeated roots. Latifa says that if such an equation has exactly one real root, it must be of degree 3 Sam says that this is not correct. State, giving reasons, whether Latifa or Sam is right. [3 marks]

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

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]

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]

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

1. show all the roots of \(f(z) = 0\) on a single Argand diagram, [7]
2. find the value of \(p\) and the value of \(q\). [3]

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]

$$f(z) = 3z^3 + pz^2 + 57z + q$$ where \(p\) and \(q\) are real constants. Given that \(3 - 2\sqrt{21}i\) is a root of the equation \(f(z) = 0\)

1. show all the roots of \(f(z) = 0\) on a single Argand diagram, [7]
2. find the value of \(p\) and the value of \(q\). [3]