Higher degree polynomial from complex roots

Given complex roots (and possibly real roots) of a cubic or higher degree polynomial with real coefficients, form the complete polynomial equation.

2 questions

OCR MEI Further Pure Core AS 2018 June Q4
4 Find a cubic equation with real coefficients, two of whose roots are \(2 - \mathrm { i }\) and 3.
SPS SPS FM Pure 2024 February Q1
  1. The plane \(x + 2 y + c z = 4\) is perpendicular to the plane \(2 x - c y + 6 z = 9\), where \(c\) is a constant. Find the value of \(c\).
  2. Find the mean value of \(\mathrm { f } ( x ) = x ^ { 2 } + 6 x\) over the interval \([ 0,3 ]\).
  3. It is given that \(1 - 3 \mathrm { i }\) is one root of the quartic equation
$$z ^ { 4 } - 2 z ^ { 3 } + p z ^ { 2 } + r z + 80 = 0$$ where \(p\) and \(r\) are real numbers.
  1. Express \(z ^ { 4 } - 2 z ^ { 3 } + p z ^ { 2 } + r z + 80\) as the product of two quadratic factors with real coefficients.
  2. Find the value of \(p\) and the value of \(r\).
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