Fully specified polynomial: verify factor and solve

Polynomial has no unknown constants; verify a given factor using the factor theorem, factorise completely, and solve f(x) = 0.

8 questions · Moderate -0.8

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OCR MEI C1 Q7
5 marks Moderate -0.8
7 Show that ( \(x - 2\) ) is a factor of \(\mathrm { f } ( x ) = x ^ { 3 } - x ^ { 2 } - 4 x + 4\).
Hence solve the equation \(x ^ { 3 } - x ^ { 2 } - 4 x + 4 = 0\).
OCR MEI C1 Q5
5 marks Moderate -0.8
5 You are given that \(\mathrm { f } ( x ) = x ^ { 3 } - 7 x + 6\).
  1. Show that ( \(x - 2\) ) is a factor of \(\mathrm { f } ( x )\).
  2. Solve the equation \(\mathrm { f } ( x ) = 0\).
OCR MEI AS Paper 1 2018 June Q6
5 marks Moderate -0.8
6 In this question you must show detailed reasoning.
You are given that \(\mathrm { f } ( x ) = 4 x ^ { 3 } - 3 x + 1\).
  1. Use the factor theorem to show that \(( x + 1 )\) is a factor of \(\mathrm { f } ( x )\).
  2. Solve the equation \(\mathrm { f } ( x ) = 0\).
OCR MEI Paper 1 2020 November Q7
6 marks Moderate -0.8
7 In this question you must show detailed reasoning. The function \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = x ^ { 3 } + x ^ { 2 } - 8 x - 12\) for all values of \(x\).
  1. Use the factor theorem to show that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\).
  2. Solve the equation \(\mathrm { f } ( x ) = 0\).
Pre-U Pre-U 9794/1 2012 June Q2
6 marks Moderate -0.8
2 Let \(\mathrm { f } ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 13 x + 15\).
  1. Show that \(\mathrm { f } ( 1 ) = 0\) and hence factorise \(x ^ { 3 } - 3 x ^ { 2 } - 13 x + 15\) completely.
  2. Hence solve the equation \(x ^ { 3 } - 3 x ^ { 2 } - 13 x + 15 = 0\).
Pre-U Pre-U 9794/1 2012 Specimen Q2
5 marks Moderate -0.8
2
  1. Show that \(x = 2\) is a root of the equation \(2 x ^ { 3 } - x ^ { 2 } - 15 x + 18 = 0\).
  2. Hence solve the equation \(2 x ^ { 3 } - x ^ { 2 } - 15 x + 18 = 0\).
Pre-U Pre-U 9794/1 2016 Specimen Q4
6 marks Moderate -0.8
4
  1. Show that \(x = 2\) is a root of the equation \(2 x ^ { 3 } - x ^ { 2 } - 15 x + 18 = 0\).
  2. Hence solve the equation \(2 x ^ { 3 } - x ^ { 2 } - 15 x + 18 = 0\).
Pre-U Pre-U 9794/1 2019 Specimen Q4
1 marks Moderate -0.8
4
  1. Show that \(x = 2\) is a root of the equation \(2 x ^ { 3 } - x ^ { 2 } - 15 x + 18 = 0\).
  2. Hence solve the equation \(2 x ^ { 3 } - x ^ { 2 } - 15 x + 18 = 0\).