Find conjugate roots from polynomial

A question is this type if and only if it asks to find other roots of a polynomial with real coefficients given one complex root, using the conjugate root theorem.

7 questions · Standard +0.5

4.02g Conjugate pairs: real coefficient polynomials
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CAIE P3 2021 June Q10
10 marks Standard +0.3
10
  1. Verify that \(- 1 + \sqrt { 2 } \mathrm { i }\) is a root of the equation \(z ^ { 4 } + 3 z ^ { 2 } + 2 z + 12 = 0\).
  2. Find the other roots of this equation.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
Edexcel F1 2023 June Q2
7 marks Standard +0.3
  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.
Edexcel F1 2018 Specimen Q8
9 marks Moderate -0.3
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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AQA Further Paper 1 2020 June Q4
6 marks Standard +0.8
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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    4
  2. Find the value of \(p\) and the value of \(r\).
SPS SPS FM 2023 January Q8
10 marks Standard +0.8
$$f(z) = 3z^3 + pz^2 + 57z + q$$ where \(p\) and \(q\) are real constants. Given that \(3 - 2\sqrt{2}i\) is a root of the equation \(f(z) = 0\)
  1. show all the roots of \(f(z) = 0\) on a single Argand diagram, [7]
  2. find the value of \(p\) and the value of \(q\). [3]
SPS SPS FM Pure 2025 February Q6
10 marks Standard +0.8
$$f(z) = 3z^3 + pz^2 + 57z + q$$ where \(p\) and \(q\) are real constants. Given that \(3 - 2\sqrt{21}i\) is a root of the equation \(f(z) = 0\)
  1. show all the roots of \(f(z) = 0\) on a single Argand diagram, [7]
  2. find the value of \(p\) and the value of \(q\). [3]
SPS SPS FM Pure 2025 February Q6
10 marks Standard +0.8
$$f(z) = 3z^3 + pz^2 + 57z + q$$ where \(p\) and \(q\) are real constants. Given that \(3 - 2\sqrt{2}i\) is a root of the equation \(f(z) = 0\)
  1. show all the roots of \(f(z) = 0\) on a single Argand diagram, [7]
  2. find the value of \(p\) and the value of \(q\). [3]