Roots of polynomial equations

A question is this type if and only if it involves verifying or finding complex roots of polynomial equations, including stating conjugate roots.

6 questions · Standard +0.4

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CAIE P3 2005 June Q3
7 marks Moderate -0.3
3
  1. Solve the equation \(z ^ { 2 } - 2 \mathrm { i } z - 5 = 0\), giving your answers in the form \(x + \mathrm { i } y\) where \(x\) and \(y\) are real.
  2. Find the modulus and argument of each root.
  3. Sketch an Argand diagram showing the points representing the roots.
Edexcel FP1 Q7
12 marks Moderate -0.8
7. The quadratic equation $$z ^ { 2 } + 10 z + 169 = 0$$ has complex roots \(z _ { 1 }\) and \(z _ { 2 }\).
  1. Find each of these roots in the form \(a + b \mathrm { i }\).
  2. Find the modulus and argument of \(z _ { 1 }\) and of \(z _ { 2 }\). Give the arguments in radians to 3 significant figures.
  3. Illustrate the two roots on a single Argand diagram.
  4. Find the value of \(\left| z _ { 1 } - z _ { 2 } \right|\).
OCR Further Pure Core AS Specimen Q7
9 marks Challenging +1.2
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.
OCR MEI Further Pure Core AS 2020 November Q3
7 marks Standard +0.3
3 In this question you must show detailed reasoning.
The roots of the equation \(x ^ { 2 } - 2 x + 4 = 0\) are \(\alpha\) and \(\beta\).
  1. Find \(\alpha\) and \(\beta\) in modulus-argument form.
  2. Hence or otherwise show that \(\alpha\) and \(\beta\) are both roots of \(x ^ { 3 } + \lambda = 0\), where \(\lambda\) is a real constant to be determined.
Edexcel CP2 2023 June Q8
11 marks Challenging +1.2
  1. Given that a cubic equation has three distinct roots that all lie on the same straight line in the complex plane,
    1. describe the possible lines the roots can lie on.
    $$f ( z ) = 8 z ^ { 3 } + b z ^ { 2 } + c z + d$$ where \(b , c\) and \(d\) are real constants.
    The roots of \(f ( z )\) are distinct and lie on a straight line in the complex plane.
    Given that one of the roots is \(\frac { 3 } { 2 } + \frac { 3 } { 2 } \mathrm { i }\)
  2. state the other two roots of \(\mathrm { f } ( \mathrm { z } )\) $$g ( z ) = z ^ { 3 } + P z ^ { 2 } + Q z + 12$$ where \(P\) and \(Q\) are real constants, has 3 distinct roots.
    The roots of \(g ( z )\) lie on a different straight line in the complex plane than the roots of \(\mathrm { f } ( \mathrm { z } )\) Given that
    • \(f ( z )\) and \(g ( z )\) have one root in common
    • one of the roots of \(\mathrm { g } ( \mathrm { z } )\) is - 4
      1. write down the value of the common root,
      2. determine the value of the other root of \(\mathrm { g } ( \mathrm { z } )\)
    • Hence solve the equation \(\mathrm { f } ( \mathrm { z } ) = \mathrm { g } ( \mathrm { z } )\)
OCR FP1 AS 2017 Specimen Q7
9 marks Standard +0.8
7 In this question you must show detailed reasoning. It is given that \(\mathrm { f } ( z ) = z ^ { 3 } - 13 z ^ { 2 } + 65 z - 125\).
The points representing the three roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\) 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.