Complex roots with real coefficients

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 \(\mathrm { p } ( x ) = 0\).

10 The complex number \(- 1 + \sqrt { 7 } \mathrm { i }\) is denoted by \(u\). It is given that \(u\) is a root of the equation $$2 x ^ { 3 } + 3 x ^ { 2 } + 14 x + k = 0$$ where \(k\) is a real constant.

1. Find the value of \(k\).
2. Find the other two roots of the equation.
3. On an Argand diagram, sketch the locus of points representing complex numbers \(z\) satisfying the equation \(| z - u | = 2\).
4. Determine the greatest value of \(\arg z\) for points on this locus, giving your answer in radians.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7

1. Verify that \(- 1 + \sqrt { 5 } \mathrm { i }\) is a root of the equation \(2 x ^ { 3 } + x ^ { 2 } + 6 x - 18 = 0\).
2. Find the other roots of this equation.

10 The complex number \(1 + 2 \mathrm { i }\) is denoted by \(u\). The polynomial \(2 x ^ { 3 } + a x ^ { 2 } + 4 x + b\), where \(a\) and \(b\) are real constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(u\) is a root of the equation \(\mathrm { p } ( x ) = 0\).

1. Find the values of \(a\) and \(b\).
2. State a second complex root of this equation.
3. Find the real factors of \(\mathrm { p } ( x )\).
4. 1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - u | \leqslant \sqrt { 5 }\) and \(\arg z \leqslant \frac { 1 } { 4 } \pi\).
   2. Find the least value of \(\operatorname { Im } z\) for points in the shaded region. Give your answer in an exact form.
      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. $$f ( x ) = x ^ { 4 } - x ^ { 3 } - 9 x ^ { 2 } + 29 x - 60$$ 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 \(\mathrm { f } ( x ) = 0\)

7. $$f ( x ) = x ^ { 4 } - 3 x ^ { 3 } - 15 x ^ { 2 } + 99 x - 130$$

1. Given that \(x = 3 + 2 \mathrm { i }\) 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. Show the four roots of \(\mathrm { f } ( x ) = 0\) on a single Argand diagram.

3. $$f ( x ) = x ^ { 4 } + 2 x ^ { 3 } + 26 x ^ { 2 } + 32 x + 160$$ Given that \(x = - 1 + 3 \mathrm { i }\) is a root of the equation \(\mathrm { f } ( x ) = 0\), use algebra to find the three other roots of \(\mathrm { f } ( x ) = 0\)
(Solutions based entirely on graphical or numerical methods are not acceptable.)

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 \(q\).

4. The equation $$x ^ { 4 } + A x ^ { 3 } + B x ^ { 2 } + C x + 225 = 0$$ where \(A , B\) and \(C\) are real constants, has

• a complex root \(4 + 3 \mathrm { i }\)
• a repeated positive real root
  1. Write down the other complex root of this equation.
  2. Hence determine a quadratic factor of \(x ^ { 4 } + A x ^ { 3 } + B x ^ { 2 } + C x + 225\)
  3. Deduce the real root of the equation.
  4. Hence determine the value of each of the constants \(A , B\) and \(C\)

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.

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

1. write down the other root of the equation.
2. Find the value of \(p\) and the value of \(q\).

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

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

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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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 \(B\).

4. $$f ( z ) = 2 z ^ { 3 } - z ^ { 2 } + a z + b$$ where \(a\) and \(b\) are integers. The complex number \(- 1 - 3 \mathrm { i }\) is a root of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)

1. Write down another complex root of this equation.
2. Determine the value of \(a\) and the value of \(b\).
3. Show all the roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\) on a single Argand diagram.

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6. Given that 2 and \(5 + 2 \mathrm { i }\) are roots of the equation $$x ^ { 3 } - 12 x ^ { 2 } + c x + d = 0 , \quad c , d \in \mathbb { R }$$

1. write down the other complex root of the equation.
2. Find the value of \(c\) and the value of \(d\).
3. Show the three roots of this equation on a single Argand diagram.

4. Given that \(2 - 4 \mathrm { i }\) is a root of the equation $$z ^ { 2 } + p z + q = 0 ,$$ where \(p\) and \(q\) are real constants,

1. write down the other root of the equation,
2. find the value of \(p\) and the value of \(q\).

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.
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1. The roots of the equation

$$2 z ^ { 3 } - 3 z ^ { 2 } + 8 z + 5 = 0$$ are \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\)
Given that \(z _ { 1 } = 1 + 2 i\), find \(z _ { 2 }\) and \(z _ { 3 }\)

3. Given that 2 and \(1 - 5 \mathrm { i }\) are roots of the equation $$x ^ { 3 } + p x ^ { 2 } + 30 x + q = 0 , \quad p , q \in \mathbb { R }$$

1. write down the third root of the equation.
2. Find the value of \(p\) and the value of \(q\).
3. Show the three roots of this equation on a single Argand diagram.

6. Given that 4 and \(2 \mathrm { i } - 3\) are roots of the equation $$x ^ { 3 } + a x ^ { 2 } + b x - 52 = 0$$ where \(a\) and \(b\) are real constants,

1. write down the third root of the equation,
2. find the value of \(a\) and the value of \(b\).

8. $$\mathrm { f } ( x ) \equiv 2 x ^ { 3 } - 5 x ^ { 2 } + p x - 5 , p \in \mathbb { R }$$ Given that \(1 - 2 \mathrm { i }\) is a complex solution of \(\mathrm { f } ( x ) = 0\),

1. write down the other complex solution of \(\mathrm { f } ( x ) = 0\),
2. solve the equation \(\mathrm { f } ( x ) = 0\),
3. find the value of \(p\).

6 The cubic equation \(x ^ { 3 } + a x ^ { 2 } + b x + c = 0\), where \(a , b\) and \(c\) are real, has roots ( \(3 + \mathrm { i }\) ) and 2 .

1. Write down the other root of the equation.
2. Find the values of \(a , b\) and \(c\).