Verify roots satisfy polynomial equations

A question is this type if and only if it asks to verify or show that a given complex number satisfies a particular polynomial equation by direct substitution.

2 questions · Standard +0.0

4.02i Quadratic equations: with complex roots
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CAIE P3 2011 June Q7
8 marks Standard +0.3
7
  1. Find the roots of the equation $$z ^ { 2 } + ( 2 \sqrt { } 3 ) z + 4 = 0$$ giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. State the modulus and argument of each root.
  3. Showing all your working, verify that each root also satisfies the equation $$z ^ { 6 } = - 64$$
Pre-U Pre-U 9794/1 Specimen Q9
5 marks Moderate -0.3
9
  1. Show that \(z = ( 1 + \mathrm { i } )\) is a root of the cubic equation \(3 z ^ { 3 } - 8 z ^ { 2 } + 10 z - 4 = 0\).
  2. Show that the equation \(3 z ^ { 3 } - 8 z ^ { 2 } + 10 z - 4 = 0\) has a quadratic factor with real coefficients and hence solve this equation completely.