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

10 Throughout this question the use of a calculator is not permitted.
The complex number \(( \sqrt { } 3 ) + \mathrm { i }\) is denoted by \(u\).

1. Express \(u\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\), giving the exact values of \(r\) and \(\theta\). Hence or otherwise state the exact values of the modulus and argument of \(u ^ { 4 }\).
2. Verify that \(u\) is a root of the equation \(z ^ { 3 } - 8 z + 8 \sqrt { } 3 = 0\) and state the other complex root of this equation.
3. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - u | \leqslant 2\) and \(\operatorname { Im } z \geqslant 2\), where \(\operatorname { Im } z\) denotes the imaginary part of \(z\). If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

1 Find the exact value of the constant \(k\) for which \(\int _ { 1 } ^ { k } \frac { 1 } { 2 x - 1 } \mathrm {~d} x = 1\).

7 The complex number \(- 2 + \mathrm { i }\) is denoted by \(u\).

1. Given that \(u\) is a root of the equation \(x ^ { 3 } - 11 x - k = 0\), where \(k\) is real, find the value of \(k\).
2. Write down the other complex root of this equation.
3. Find the modulus and argument of \(u\).
4. Sketch an Argand diagram showing the point representing \(u\). Shade the region whose points represent the complex numbers \(z\) satisfying both the inequalities $$| z | < | z - 2 | \quad \text { and } \quad 0 < \arg ( z - u ) < \frac { 1 } { 4 } \pi$$

9 The complex number \(1 + ( \sqrt { } 2 ) \mathrm { i }\) is denoted by \(u\). The polynomial \(x ^ { 4 } + x ^ { 2 } + 2 x + 6\) is denoted by \(\mathrm { p } ( x )\).

1. Showing your working, verify that \(u\) is a root of the equation \(\mathrm { p } ( x ) = 0\), and write down a second complex root of the equation.
2. Find the other two roots of the equation \(\mathrm { p } ( x ) = 0\).

1

1. Find \(a\) and \(b\) such that $$z ^ { 8 } - i z ^ { 5 } - z ^ { 3 } + i = \left( z ^ { 5 } - a \right) \left( z ^ { 3 } - b \right) .$$
2. Hence find the roots of $$z ^ { 8 } - i z ^ { 5 } - z ^ { 3 } + i = 0$$ giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(0 \leqslant \theta < 2 \pi\).

4. $$f ( x ) = x ^ { 4 } + 3 x ^ { 3 } - 5 x ^ { 2 } - 19 x - 60$$

1. Given that \(x = - 4\) and \(x = 3\) are roots of the equation \(\mathrm { f } ( x ) = 0\), use algebra to solve \(\mathrm { f } ( x ) = 0\) completely.
2. Show the four roots of \(\mathrm { f } ( x ) = 0\) on a single Argand diagram.

1. $$f ( x ) = x ^ { 4 } - x ^ { 3 } - 9 x ^ { 2 } + 29 x - 60$$ Given that \(x = 1 + 2 \mathrm { i }\) is a root of the equation \(\mathrm { f } ( x ) = 0\), use algebra to find the three other roots of the equation \(\mathrm { f } ( x ) = 0\)

2. $$f ( z ) = z ^ { 4 } - 6 z ^ { 3 } + 38 z ^ { 2 } - 94 z + 221$$

1. Given that \(z = 2 + 3 i\) is a root of the equation \(f ( z ) = 0\), use algebra to find the three other roots of \(f ( z ) = 0\)
2. Show the four roots of \(\mathrm { f } ( \mathrm { z } ) = 0\) on a single Argand diagram.

1. Given that \(x = \frac { 3 } { 8 } + \frac { \sqrt { 71 } } { 8 } \mathrm { i }\) is a root of the equation

$$4 x ^ { 3 } - 19 x ^ { 2 } + p x + q = 0$$

1. write down the other complex root of the equation. Given that \(x = 4\) is also a root of the equation,
2. find the value of \(p\) and the value of \(q\).

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

$$\mathrm { f } ( z ) = 4 z ^ { 3 } + p z ^ { 2 } - 24 z + 108$$ where \(p\) is a constant.
Given that - 3 is a root of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)

1. determine the value of \(p\)
2. using algebra, solve \(\mathrm { f } ( \mathrm { z } ) = 0\) completely, giving the roots in simplest form,
3. determine the modulus of the complex roots of \(\mathrm { f } ( \mathrm { z } ) = 0\)
4. show the roots of \(\mathrm { f } ( \mathrm { z } ) = 0\) on a single Argand diagram.

2. $$f ( z ) = 2 z ^ { 3 } + p z ^ { 2 } + q z - 41$$ where \(p\) and \(q\) are integers.
The complex number \(5 - 4 \mathrm { i }\) is a root of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)

1. Write down another complex root of this equation.
2. Solve the equation \(\mathrm { f } ( \mathrm { z } ) = 0\) completely.
3. Determine the value of \(p\) and the value of \(q\). When plotted on an Argand diagram, the points representing the roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\) form the vertices of a triangle.
4. Determine the area of this triangle.

7. $$f ( z ) = z ^ { 4 } + 4 z ^ { 3 } + 6 z ^ { 2 } + 4 z + a$$ where \(a\) is a real constant. Given that \(1 + 2 \mathrm { i }\) is a complex root of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)

1. write down another complex root of this equation.
2. 1. Hence, find the other roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)
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5. Given that $$z ^ { 4 } - 6 z ^ { 3 } + 34 z ^ { 2 } - 54 z + 225 \equiv \left( z ^ { 2 } + 9 \right) \left( z ^ { 2 } + a z + b \right)$$ where \(a\) and \(b\) are real numbers,

1. find the value of \(a\) and the value of \(b\).
2. Hence find the exact roots of the equation $$z ^ { 4 } - 6 z ^ { 3 } + 34 z ^ { 2 } - 54 z + 225 = 0$$
3. Show your roots on a single Argand diagram.

3. $$f ( z ) = z ^ { 4 } + a z ^ { 3 } + b z ^ { 2 } + c z + d$$ where \(a , b , c\) and \(d\) are integers.
The complex numbers \(3 + \mathrm { i }\) and \(- 1 - 2 \mathrm { i }\) are roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)

1. Write down the other roots of this equation.
2. Show all the roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\) on a single Argand diagram.
3. Determine the values of \(a , b , c\) and \(d\).

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4. $$f ( z ) = 2 z ^ { 4 } - 19 z ^ { 3 } + A z ^ { 2 } + B z - 156$$ where \(A\) and \(B\) are constants.
The complex number \(5 - \mathrm { i }\) is a root of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)

1. Write down another complex root of this equation.
2. Solve the equation \(\mathrm { f } ( \mathrm { z } ) = 0\) completely.
3. Determine the value of \(A\) and the value of \(B\).

1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

Given that \(x = 2 + 3 \mathrm { i }\) is a root of the equation $$2 x ^ { 4 } - 8 x ^ { 3 } + 29 x ^ { 2 } - 12 x + 39 = 0$$

1. write down another complex root of this equation.
2. Use algebra to determine the other 2 roots of the equation.
3. Show all 4 roots on a single Argand diagram.

1. In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable. $$\mathrm { f } ( z ) = z ^ { 3 } - 13 z ^ { 2 } + 59 z + p \quad p \in \mathbb { Z }$$ Given that \(z = 3\) is a root of the equation \(f ( z ) = 0\)

1. show that \(p = - 87\)
2. Use algebra to determine the other roots of \(\mathrm { f } ( \mathrm { z } ) = 0\), giving your answers in simplest form. On an Argand diagram
   • the root \(z = 3\) is represented by the point \(P\)
   • the other roots of \(\mathrm { f } ( \mathrm { z } ) = 0\) are represented by the points \(Q\) and \(R\)
   • the number \(z = - 9\) is represented by the point \(S\)
   • Show on a single Argand diagram the positions of \(P , Q , R\) and \(S\)
   • Determine the perimeter of the quadrilateral \(P Q S R\), giving your answer as a simplified surd.

8. $$\mathrm { f } ( z ) = z ^ { 4 } + 6 z ^ { 3 } + 76 z ^ { 2 } + a z + b$$ where \(a\) and \(b\) are real constants.
Given that \(- 3 + 8 \mathrm { i }\) is a complex root of the equation \(\mathrm { f } ( z ) = 0\)

1. write down another complex root of this equation.
2. Hence, or otherwise, find the other roots of the equation \(\mathrm { f } ( z ) = 0\)
3. Show on a single Argand diagram all four roots of the equation \(\mathrm { f } ( z ) = 0\)

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2. $$f ( z ) = z ^ { 3 } + 5 z ^ { 2 } + 11 z + 15$$ Given that \(z = 2 i - 1\) is a solution of the equation \(f ( z ) = 0\), use algebra to solve \(f ( z ) = 0\) completely.
(5)

1. (a) Use de Moivre's theorem to show that

$$\tan 4 \theta \equiv \frac { 4 \tan \theta - 4 \tan ^ { 3 } \theta } { 1 - 6 \tan ^ { 2 } \theta + \tan ^ { 4 } \theta }$$ (b) Use the identity given in part (a) to find the 2 positive roots of $$x ^ { 4 } + 2 x ^ { 3 } - 6 x ^ { 2 } - 2 x + 1 = 0$$ giving your answers to 3 significant figures. \includegraphics[max width=\textwidth, alt={}, center]{0d44aec7-a6e8-47fc-a215-7c8c4790e93f-29_2255_50_314_35}

3

1. Show that \(z = 3\) is a root of the cubic equation \(z ^ { 3 } + z ^ { 2 } - 7 z - 15 = 0\) and find the other roots.
2. Show the roots on an Argand diagram.

8 The function \(\mathrm { f } ( z ) = z ^ { 4 } - z ^ { 3 } + a z ^ { 2 } + b z + c\) has real coefficients. The equation \(\mathrm { f } ( z ) = 0\) has roots \(\alpha , \beta\), \(\gamma\) and \(\delta\) where \(\alpha = 1\) and \(\beta = 1 + \mathrm { j }\).

1. Write down the other complex root and explain why the equation must have a second real root.
2. Write down the value of \(\alpha + \beta + \gamma + \delta\) and find the second real root.
3. Find the values of \(a , b\) and \(c\).
4. Write down \(\mathrm { f } ( - z )\) and the roots of \(\mathrm { f } ( - z ) = 0\).

3 You are given that \(z = 2 + \mathrm { j }\) is a root of the cubic equation \(2 z ^ { 3 } + p z ^ { 2 } + 22 z - 15 = 0\), where \(p\) is real. Find the other roots and the value of \(p\).

3 The cubic equation \(2 z ^ { 3 } - z ^ { 2 } + 4 z + k = 0\), where \(k\) is real, has a root \(z = 1 + 2 \mathrm { j }\).
Write down the other complex root. Hence find the real root and the value of \(k\).

8 A polynomial \(\mathrm { P } ( z )\) has real coefficients. Two of the roots of \(\mathrm { P } ( z ) = 0\) are \(2 - \mathrm { j }\) and \(- 1 + 2 \mathrm { j }\).

1. Explain why \(\mathrm { P } ( z )\) cannot be a cubic. You are given that \(\mathrm { P } ( z )\) is a quartic.
2. Write down the other roots of \(\mathrm { P } ( z ) = 0\) and hence find \(\mathrm { P } ( z )\) in the form \(z ^ { 4 } + a z ^ { 3 } + b z ^ { 2 } + c z + d\).
3. Show the roots of \(\mathrm { P } ( z ) = 0\) on an Argand diagram and give, in terms of \(z\), the equation of the circle they lie on.