4.02g Conjugate pairs: real coefficient polynomials

5. The roots of the equation $$z ^ { 3 } - 8 z ^ { 2 } + 22 z - 20 = 0$$ are \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\).

1. Given that \(z _ { 1 } = 3 + \mathrm { i }\), find \(z _ { 2 }\) and \(z _ { 3 }\).
2. Show, on a single Argand diagram, the points representing \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\).

5. $$f ( x ) = \left( 4 x ^ { 2 } + 9 \right) \left( x ^ { 2 } - 6 x + 34 \right)$$

1. Find the four roots of \(\mathrm { f } ( x ) = 0\) Give your answers in the form \(x = p + \mathrm { i } q\), where \(p\) and \(q\) are real.
2. Show these four roots on a single Argand diagram.

3. $$f ( x ) = \left( x ^ { 2 } + 4 \right) \left( x ^ { 2 } + 8 x + 25 \right)$$

1. Find the four roots of \(\mathrm { f } ( x ) = 0\).
2. Find the sum of these four roots.

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.

4. $$z = \frac { 4 } { 1 + \mathrm { i } }$$ Find, in the form \(a + \mathrm { i } b\) where \(a , b \in \mathbb { R }\)

1. \(Z\)
2. \(z ^ { 2 }\) Given that \(z\) is a complex root of the quadratic equation \(x ^ { 2 } + p x + q = 0\), where \(p\) and \(q\) are real integers,
3. find the value of \(p\) and the value of \(q\).

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 \(p\).

3 One root of the quadratic equation \(x ^ { 2 } + p x + q = 0\), where \(p\) and \(q\) are real, is the complex number 2-3i.

1. Write down the other root.
2. Find the values of \(p\) and \(q\).

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 \(c\).

8 You are given that the complex number \(\alpha = 1 + \mathrm { j }\) satisfies the equation \(z ^ { 3 } + 3 z ^ { 2 } + p z + q = 0\), where \(p\) and \(q\) are real constants.

1. Find \(\alpha ^ { 2 }\) and \(\alpha ^ { 3 }\) in the form \(a + b \mathrm { j }\). Hence show that \(p = - 8\) and \(q = 10\).
2. Find the other two roots of the equation.
3. Represent the three roots on an Argand diagram.

9 The quartic equation \(x ^ { 4 } + A x ^ { 3 } + B x ^ { 2 } + C x + D = 0\), where \(A , B , C\) and \(D\) are real numbers, has roots \(2 + \mathrm { j }\) and - 2 j .

1. Write down the other roots of the equation.
2. Find the values of \(A , B , C\) and \(D\).

9 Two complex numbers, \(\alpha\) and \(\beta\), are given by \(\alpha = 2 - 2 \mathrm { j }\) and \(\beta = - 1 + \mathrm { j }\). \(\alpha\) and \(\beta\) are both roots of a quartic equation \(x ^ { 4 } + A x ^ { 3 } + B x ^ { 2 } + C x + D = 0\), where \(A , B , C\) and \(D\) are real numbers.

1. Write down the other two roots.
2. Represent these four roots on an Argand diagram.
3. Find the values of \(A , B , C\) and \(D\).

6 One root of the cubic equation \(x ^ { 3 } + p x ^ { 2 } + 6 x + q = 0\), where \(p\) and \(q\) are real, is the complex number 5-i.

1. Find the real root of the cubic equation.
2. Find the values of \(p\) and \(q\).

9 One root of the quadratic equation \(x ^ { 2 } + a x + b = 0\), where \(a\) and \(b\) are real, is \(16 - 30 \mathrm { i }\).

1. Write down the other root of the quadratic equation.
2. Find the values of \(a\) and \(b\).
3. Use an algebraic method to solve the quartic equation \(y ^ { 4 } + a y ^ { 2 } + b = 0\).

3 One root of the quadratic equation \(x ^ { 2 } + a x + b = 0\), where \(a\) and \(b\) are real, is the complex number \(4 - 3 \mathrm { i }\). Find the values of \(a\) and \(b\).

9 The roots of the equation \(x ^ { 3 } - k x ^ { 2 } - 2 = 0\) are \(\alpha , \beta\) and \(\gamma\), where \(\alpha\) is real and \(\beta\) and \(\gamma\) are complex.

1. Show that \(k = \alpha - \frac { 2 } { \alpha ^ { 2 } }\).
2. Given that \(\beta = u + \mathrm { i } v\), where \(u\) and \(v\) are real, find \(u\) in terms of \(\alpha\).
3. Find \(v ^ { 2 }\) in terms of \(\alpha\).

9 Two complex numbers, \(\alpha\) and \(\beta\), are given by \(\alpha = 1 + \mathrm { j }\) and \(\beta = 2 - \mathrm { j }\).

1. Express \(\alpha + \beta , \alpha \alpha ^ { * }\) and \(\frac { \alpha + \beta } { \alpha }\) in the form \(a + b \mathrm { j }\).
2. Find a quadratic equation with roots \(\alpha\) and \(\alpha ^ { * }\).
3. \(\alpha\) and \(\beta\) are roots of a quartic equation with real coefficients. Write down the two other roots and find this quartic equation in the form \(z ^ { 4 } + A z ^ { 3 } + B z ^ { 2 } + C z + D = 0\).

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.

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.

7 The function \(\mathrm { f } ( z ) = 2 z ^ { 4 } - 9 z ^ { 3 } + A z ^ { 2 } + B z - 26\) has real coefficients. The equation \(\mathrm { f } ( z ) = 0\) has two real roots, \(\alpha\) and \(\beta\), where \(\alpha > \beta\), and two complex roots, \(\gamma\) and \(\delta\), where \(\gamma = 3 + 2 \mathrm { j }\).

1. Show that \(\alpha + \beta = - \frac { 3 } { 2 }\) and find the value of \(\alpha \beta\).
2. Hence find the two real roots \(\alpha\) and \(\beta\).
3. Find the values of \(A\) and \(B\).
4. Write down the roots of the equation \(\mathrm { f } \left( \frac { w } { \mathrm { j } } \right) = 0\).