4.02j Cubic/quartic equations: conjugate pairs and factor theorem

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]

In this question, \(w\) denotes the complex number \(\cos \frac{2\pi}{5} + i \sin \frac{2\pi}{5}\).

1. Express \(w^2\), \(w^3\) and \(w^4\) in polar form, with arguments in the interval \(0 \leq \theta < 2\pi\). [4]
2. The points in an Argand diagram which represent the numbers $$1, \quad 1 + w, \quad 1 + w + w^2, \quad 1 + w + w^2 + w^3, \quad 1 + w + w^2 + w^3 + w^4$$ are denoted by \(A\), \(B\), \(C\), \(D\), \(E\) respectively. Sketch the Argand diagram to show these points and join them in the order stated. (Your diagram need not be exactly to scale, but it should show the important features.) [4]
3. Write down a polynomial equation of degree 5 which is satisfied by \(w\). [1]

The equation \(z^3 + kz^2 + 9 = 0\) has roots \(\alpha\), \(\beta\) and \(\gamma\).

1. 1. Show that $$\alpha^2 + \beta^2 + \gamma^2 = k^2$$ [3 marks]
   2. Show that $$\alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2 = -18k$$ [4 marks]
2. The equation \(9z^3 - 40z^2 + rz + s = 0\) has roots \(\alpha\beta + \gamma\), \(\beta\gamma + \alpha\) and \(\gamma\alpha + \beta\).
   1. Show that $$k = -\frac{40}{9}$$ [1 mark]
   2. Without calculating the values of \(\alpha\), \(\beta\) and \(\gamma\), find the value of \(s\). Show working to justify your answer. [6 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]

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]

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]

$$f(z) = z^3 - 8z^2 + pz - 24$$ where \(p\) is a real constant. Given that the equation \(f(z) = 0\) has distinct roots $$\alpha, \beta \text{ and } \left(\alpha + \frac{12}{\alpha} - \beta\right)$$

1. solve completely the equation \(f(z) = 0\) [6]
2. Hence find the value of \(p\). [2]

$$f(z) = 3z^3 + pz^2 + 57z + q$$ 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.

1. Express \((2 + 3i)^3\) in the form \(a + ib\). [3]
2. Hence verify that \(2 + 3i\) is a root of the equation \(3z^3 - 8z^2 + 23z + 52 = 0\). [3]
3. Express \(3z^3 - 8z^2 + 23z + 52\) as the product of a linear factor and a quadratic factor with real coefficients. [4]

**In this question you must show detailed reasoning.** It is given that \(f(z) = z^3 - 13z^2 + 65z - 125\). The points representing the three roots of the equation \(f(z) = 0\) are plotted on an Argand diagram. Show that these points lie on the circle \(|z| = k\), where \(k\) is a real number to be determined. [9]

11 Answer only one of the following two alternatives. EITHER State the fifth roots of unity in the form \(\cos \theta + \mathrm { i } \sin \theta\), where \(- \pi < \theta \leqslant \pi\). Simplify $$\left( x - \left[ \cos \frac { 2 } { 5 } \pi + i \sin \frac { 2 } { 5 } \pi \right] \right) \left( x - \left[ \cos \frac { 2 } { 5 } \pi - i \sin \frac { 2 } { 5 } \pi \right] \right) .$$ Hence find the real factors of $$x ^ { 5 } - 1$$ Express the six roots of the equation $$x ^ { 6 } - x ^ { 3 } + 1 = 0$$ as three conjugate pairs, in the form \(\cos \theta \pm \mathrm { i } \sin \theta\). Hence find the real factors of $$x ^ { 6 } - x ^ { 3 } + 1$$ OR Given that $$y ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 6 y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + 3 y ^ { 3 } = 25 \mathrm { e } ^ { - 2 x }$$ and that \(v = y ^ { 3 }\), show that $$\frac { \mathrm { d } ^ { 2 } v } { \mathrm {~d} x ^ { 2 } } - 6 \frac { \mathrm {~d} v } { \mathrm {~d} x } + 9 v = 75 \mathrm { e } ^ { - 2 x }$$ Find the particular solution for \(y\) in terms of \(x\), given that when \(x = 0 , y = 2\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\). \end{document}