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

8

1. Verify that \(1 + 3 \mathrm { j }\) is a root of the equation \(3 z ^ { 3 } - 2 z ^ { 2 } + 22 z + 40 = 0\), showing your working.
2. Explain why the equation must have exactly one real root.
3. Find the other roots of the equation.

3 You are given that \(z = 2 + 3 \mathrm { j }\) is a root of the quartic equation \(z ^ { 4 } - 5 z ^ { 3 } + 15 z ^ { 2 } - 5 z - 26 = 0\). Find the other roots.

8 The complex number \(5 + 4 \mathrm { j }\) is denoted by \(\alpha\).

1. Find \(\alpha ^ { 2 }\) and \(\alpha ^ { 3 }\), showing your working.
2. The real numbers \(q\) and \(r\) are such that \(\alpha ^ { 3 } + \mathrm { q } \alpha ^ { 2 } + 11 \alpha + \mathrm { r } = 0\). Find \(q\) and \(r\). Let \(\mathrm { f } ( \mathrm { z } ) = \mathrm { z } ^ { 3 } + \mathrm { qz } ^ { 2 } + 11 \mathrm { z } + \mathrm { r }\), where \(q\) and \(r\) are as in part (ii).
3. Solve the equation \(\mathrm { f } ( z ) = 0\).
4. Solve the equation \(z ^ { 4 } + q z ^ { 3 } + 11 z ^ { 2 } + r z = z ^ { 3 } + q z ^ { 2 } + 11 z + r\).

9 A cubic equation \(x ^ { 3 } + b x ^ { 2 } + c x + d = 0\) has real roots \(\alpha , \beta\) and \(\gamma\) such that $$\begin{aligned} \frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma } & = - \frac { 5 } { 12 } \\ \alpha \beta \gamma & = - 12 \\ \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } & = 90 \end{aligned}$$

1. Find the values of \(c\) and \(d\).
2. Express \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\) in terms of \(b\).
3. Show that \(b ^ { 3 } - 15 b + 126 = 0\).
4. Given that \(3 + \mathrm { i } \sqrt { } ( 12 )\) is a root of \(y ^ { 3 } - 15 y + 126 = 0\), deduce the value of \(b\).

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 $$x ^ { 6 } - x ^ { 3 } + 1$$

11 A point has Cartesian coordinates \(( x , y )\) and polar coordinates \(( r , \theta )\) where \(r \geq 0\) and \(- \pi < \theta \leq \pi\) 11

1. Express \(r\) in terms of \(x\) and \(y\) 11
2. Express \(x\) in terms of \(r\) and \(\theta\) 11
3. The curve \(C _ { 1 }\) has the polar equation $$r ( 2 + \cos \theta ) = 1 \quad - \pi < \theta \leq \pi$$ 11 (c) (i) Show that the Cartesian equation of \(C _ { 1 }\) can be written as $$a y ^ { 2 } = ( 1 + b x ) ( 1 + x )$$ where \(a\) and \(b\) are integers to be determined.
   11 (c) (ii) The curve \(C _ { 2 }\) has the Cartesian equation $$a x ^ { 2 } = ( 1 + b y ) ( 1 + y )$$ where \(a\) and \(b\) take the same values as in part (c)(i). Describe fully a single transformation that maps the curve \(C _ { 1 }\) onto the curve \(C _ { 2 }\)

5 In this question you must show detailed reasoning.

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

7 In this question you must show detailed reasoning.
It is given that \(\mathrm { f } ( \mathrm { z } ) = \mathrm { z } ^ { 3 } - 13 z ^ { 2 } + 65 z - 125\).
The points representing the three roots of the equation \(\mathrm { f } ( z ) = 0\) 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.

3 In this question you must show detailed reasoning.
You are given that \(x = 2 + 5 \mathrm { i }\) is a root of the equation \(x ^ { 3 } - 2 x ^ { 2 } + 21 x + 58 = 0\).
Solve the equation.

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

1. Express \(\mathrm { f } ( x )\) as a product of two quadratic factors.
2. Hence write down all the roots of the equation \(\mathrm { f } ( x ) = 0\).

3. Given that \(5 - \mathrm { i }\) is a root of the equation \(x ^ { 4 } - 10 x ^ { 3 } + 10 x ^ { 2 } + 160 x - 416 = 0\),

1. write down another root of the equation,
2. find the remaining roots.

7. $$f ( z ) = z ^ { 4 } - 6 z ^ { 3 } + p z ^ { 2 } + q z + r$$ where \(p , q\) and \(r\) are real constants.
The roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\) are \(\alpha , \beta , \gamma\) and \(\delta\) where \(\alpha = 3\) and \(\beta = 2 + \mathrm { i }\) Given that \(\gamma\) is a complex root of \(\mathrm { f } ( \mathrm { z } ) = 0\)

1. 1. write down the root \(\gamma\),
   2. explain why \(\delta\) must be real.
2. Determine the value of \(\delta\).
3. Hence determine the values of \(p , q\) and \(r\).
4. Write down the roots of the equation \(\mathrm { f } ( - 2 \mathrm { z } ) = 0\)

1. $$f ( z ) = z ^ { 4 } + a z ^ { 3 } + b z ^ { 2 } + c z + d$$ where \(a , b , c\) and \(d\) are real constants.
Given that \(- 1 + 2 \mathrm { i }\) and \(3 - \mathrm { i }\) are two roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)

1. show all the roots of \(f ( z ) = 0\) on a single Argand diagram,
2. find the values of \(a , b , c\) and \(d\).

5 Throughout this question the use of a calculator is not permitted. It is given that the complex number \(- 1 + ( \sqrt { } 3 ) \mathrm { i }\) is a root of the equation $$k x ^ { 3 } + 5 x ^ { 2 } + 10 x + 4 = 0$$ where \(k\) is a real constant.

1. Write down another root of the equation.
2. Find the value of \(k\) and the third root of the equation.

10 The Argand diagram below shows the origin \(O\) and pentagon \(A B C D E\), where \(A , B , C , D\) and \(E\) are the points that represent the complex numbers \(a , b , c , d\) and \(e\), and where \(a\) is a positive real number. You are given that these five complex numbers are the roots of the equation \(z ^ { 5 } - a ^ { 5 } = 0\). \includegraphics[max width=\textwidth, alt={}, center]{bc258133-b0d6-49bb-96a7-a5ef7f9c31fc-04_885_851_482_516}

1. Justify each of the following statements.
   1. \(A , B , C , D\) and \(E\) lie on a circle with centre \(O\).
   2. \(A B C D E\) is a regular pentagon.
   3. \(b \times \mathrm { e } ^ { \frac { 2 \mathrm { i } \pi } { 5 } } = c\)
   4. \(b ^ { * } = e\)
   5. \(a + b + c + d + e = 0\)
   6. The midpoints of sides \(A B , B C , C D , D E\) and \(E A\) represent the complex numbers \(p , q , r , s\) and \(t\). Determine a polynomial equation, with real coefficients, that has roots \(p , q , r , s\) and \(t\).

6 It is given that \(z = \mathrm { e } ^ { \mathrm { i } \theta }\).

1. 1. Show that $$z + \frac { 1 } { z } = 2 \cos \theta$$
   2. Find a similar expression for $$z ^ { 2 } + \frac { 1 } { z ^ { 2 } }$$ (2 marks)
   3. Hence show that $$z ^ { 2 } - z + 2 - \frac { 1 } { z } + \frac { 1 } { z ^ { 2 } } = 4 \cos ^ { 2 } \theta - 2 \cos \theta$$ (3 marks)
2. Hence solve the quartic equation $$z ^ { 4 } - z ^ { 3 } + 2 z ^ { 2 } - z + 1 = 0$$ giving the roots in the form \(a + \mathrm { i } b\).

8 Stephen is correctly told that \(( 1 + \mathrm { i } )\) and - 1 are two roots of the polynomial equation $$z ^ { 3 } - 2 \mathrm { i } z ^ { 2 } + p z + q = 0$$ where \(p\) and \(q\) are complex numbers.
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1. Stephen states that ( \(1 - \mathrm { i }\) ) must also be a root of the equation because roots of polynomial equations occur in conjugate pairs. Explain why Stephen's reasoning is wrong. 8
2. \(\quad\) Find \(p\) and \(q\)

4

1. 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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2. Find the value of \(p\) and the value of \(r\).

4 In this question you must show detailed reasoning. You are given that \(\mathrm { f } ( \mathrm { z } ) = 4 \mathrm { z } ^ { 4 } - 12 \mathrm { z } ^ { 3 } + 41 \mathrm { z } ^ { 2 } - 128 \mathrm { z } + 185\) and that \(2 + \mathrm { i }\) is a root of the equation \(f ( z ) = 0\).

1. Express \(\mathrm { f } ( \mathrm { z } )\) as the product of two quadratic factors with integer coefficients.
2. Solve \(f ( z ) = 0\). Two loci on an Argand diagram are defined by \(C _ { 1 } = \left\{ z : | z | = r _ { 1 } \right\}\) and \(C _ { 2 } = \left\{ z : | z | = r _ { 2 } \right\}\) where \(r _ { 1 } > r _ { 2 }\). You are given that two of the points representing the roots of \(\mathrm { f } ( \mathrm { z } ) = 0\) 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. \(\omega\) is the sum of all the roots of \(\mathrm { f } ( \mathrm { z } ) = 0\). 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. You are given that \(\mathrm { f } ( z ) = 4 z ^ { 4 } - 12 z ^ { 3 } + 41 z ^ { 2 } - 128 z + 185\) and that \(2 + \mathrm { i }\) is a root of the equation \(\mathrm { f } ( z ) = 0\).

1. Express \(\mathrm { f } ( z )\) as the product of two quadratic factors with integer coefficients.
2. Solve \(\mathrm { f } ( z ) = 0\). Two loci on an Argand diagram are defined by \(C _ { 1 } = \left\{ z : | z | = r _ { 1 } \right\}\) and \(C _ { 2 } = \left\{ z : | z | = r _ { 2 } \right\}\) where \(r _ { 1 } > r _ { 2 }\). You are given that two of the points representing the roots of \(\mathrm { f } ( z ) = 0\) are on \(C _ { 1 }\) and two are on \(C _ { 2 } \cdot R\) is the region on the Argand diagram between \(C _ { 1 }\) and \(C _ { 2 }\).
3. Find the exact area of \(R\).
4. \(\omega\) is the sum of all the roots of \(\mathrm { f } ( z ) = 0\). Determine whether or not the point on the Argand diagram which represents \(\omega\) lies in \(R\).

11

1. Determine \(p\) and \(q\) given that \(( p + \mathrm { i } q ) ^ { 2 } = 63 - 16 \mathrm { i }\) and that \(p\) and \(q\) are real.
2. Let \(\mathrm { f } ( z ) = z ^ { 3 } - A z ^ { 2 } + B z - C\) for complex numbers \(A , B\) and \(C\).
   1. Given that the cubic equation \(\mathrm { f } ( z ) = 0\) has roots \(\alpha = - 7 \mathrm { i } , \beta = 3 \mathrm { i }\) and \(\gamma = 4\), determine each of \(A , B\) and \(C\).
   2. Find the roots of the equation \(\mathrm { f } ^ { \prime } ( z ) = 0\).

7

1. Express \(z ^ { 4 } + 3 z ^ { 2 } - 4\) in the form \(\left( z ^ { 2 } + a \right) \left( z ^ { 2 } + b \right)\) where \(a\) and \(b\) are real constants to be found.
2. Hence draw an Argand diagram showing the points that represent the roots of the equation \(z ^ { 4 } + 3 z ^ { 2 } - 4 = 0\).

The complex number \(z\), where \(0 < \arg z < \frac{1}{2}\pi\), is such that \(z^2 = 3 + 4\text{i}\).

1. Use an algebraic method to find \(z\). [5]
2. Show that \(z^3 = 2 + 11\text{i}\). [1]

The complex number \(w\) is the root of the equation $$w^6 - 4w^3 + 125 = 0$$ for which \(-\frac{1}{2}\pi < \arg w < 0\).

1. Find \(w\). [5]