Given a real root, find complex roots of cubic

Given a real root (or asked to verify one) of a cubic polynomial, find the remaining two complex roots by factoring out the known root and solving the resulting quadratic.

3 questions · Standard +0.1

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Pre-U Pre-U 9794/2 2013 June Q4
10 marks Moderate -0.3
4
  1. Verify that \(z = - 1\) is a root of the equation \(z ^ { 3 } + 5 z ^ { 2 } + 9 z + 5 = 0\).
  2. Find the two complex roots of the equation \(z ^ { 3 } + 5 z ^ { 2 } + 9 z + 5 = 0\).
  3. Show all three roots on an Argand diagram.
Pre-U Pre-U 9794/1 2017 June Q9
9 marks Standard +0.3
9 Solve the equation \(z ^ { 3 } + 6 z - 20 = 0\). Find the modulus and argument of each root and illustrate the roots on an Argand diagram.
Edexcel FP1 Q22
6 marks Standard +0.3
Given that \(-2\) is a root of the equation \(z^3 + 6z + 20 = 0\),
  1. Find the other two roots of the equation, [3]
  2. show, on a single Argand diagram, the three points representing the roots of the equation, [1]
  3. prove that these three points are the vertices of a right-angled triangle. [2]