Find remainder(s) then factorise

Polynomial is fully specified; find remainder(s) for given divisor(s), then use this to factorise completely.

11 questions · Moderate -0.8

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Edexcel C12 2014 January Q3
7 marks Moderate -0.8
3. $$f ( x ) = 10 x ^ { 3 } + 27 x ^ { 2 } - 13 x - 12$$
  1. Find the remainder when \(\mathrm { f } ( x )\) is divided by
    1. \(x - 2\)
    2. \(x + 3\)
  2. Hence factorise \(\mathrm { f } ( x )\) completely.
Edexcel C12 2016 October Q4
8 marks Moderate -0.8
4. $$f ( x ) = 6 x ^ { 3 } - 7 x ^ { 2 } - 43 x + 30$$
  1. Find the remainder when \(\mathrm { f } ( x )\) is divided by
    1. \(2 x + 1\)
    2. \(x - 3\)
  2. Hence factorise \(\mathrm { f } ( x )\) completely.
Edexcel C2 2011 June Q1
8 marks Moderate -0.8
1. $$f ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } - 5 x + 4$$
  1. Find the remainder when \(\mathrm { f } ( x )\) is divided by \(( x - 1 )\).
  2. Use the factor theorem to show that ( \(x + 1\) ) is a factor of \(\mathrm { f } ( x )\).
  3. Factorise f(x) completely.
Edexcel C2 2006 June Q4
8 marks Moderate -0.8
$$f ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } - 29 x - 60$$
  1. Find the remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\).
  2. Use the factor theorem to show that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\).
  3. Factorise \(\mathrm { f } ( x )\) completely.
OCR C2 2014 June Q7
9 marks Moderate -0.8
7 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 12 - 22 x + 9 x ^ { 2 } - x ^ { 3 }\).
  1. Find the remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\).
  2. Show that ( \(3 - x\) ) is a factor of \(\mathrm { f } ( x )\).
  3. Express \(\mathrm { f } ( x )\) as the product of a linear factor and a quadratic factor.
  4. Hence solve the equation \(\mathrm { f } ( x ) = 0\), giving each root in simplified surd form where appropriate.
Edexcel C2 Q2
6 marks Moderate -0.8
\(f(x) = 3x^3 - 5x^2 - 16x + 12\).
  1. Find the remainder when \(f(x)\) is divided by \((x - 2)\). [2]
Given that \((x + 2)\) is a factor of \(f(x)\),
  1. factorise \(f(x)\) completely. [4]
Edexcel C2 Q1
7 marks Moderate -0.8
  1. Find the remainder when \(x^3 - 2x^2 - 4x + 8\) is divided by
    1. \(x - 3\),
    2. \(x + 2\). [3]
  2. Hence, or otherwise, find all the solutions to the equation \(x^3 - 2x^2 - 4x + 8 = 0\). [4]
Edexcel C2 2008 January Q1
7 marks Moderate -0.8
  1. Find the remainder when $$x^3 - 2x^2 - 4x + 8$$ is divided by
    1. \(x - 3\),
    2. \(x + 2\).
    [3]
  2. Hence, or otherwise, find all the solutions to the equation $$x^3 - 2x^2 - 4x + 8 = 0.$$ [4]
OCR C2 2007 January Q8
9 marks Moderate -0.3
The polynomial f(x) is defined by \(f(x) = x^3 - 9x^2 + 7x + 33\).
  1. Find the remainder when f(x) is divided by \((x + 2)\). [2]
  2. Show that \((x - 3)\) is a factor of f(x). [1]
  3. Solve the equation f(x) = 0, giving each root in an exact form as simply as possible. [6]
Edexcel C2 Q6
9 marks Moderate -0.3
\(f(x) = 2x^3 + 3x^2 - 6x + 1\).
  1. Find the remainder when \(f(x)\) is divided by \((2x - 1)\). [2]
    1. Find the remainder when \(f(x)\) is divided by \((x + 2)\).
    2. Hence, or otherwise, solve the equation $$2x^3 + 3x^2 - 6x - 8 = 0,$$ giving your answers to 2 decimal places where appropriate. [7]
Pre-U Pre-U 9794/2 2016 June Q1
3 marks Easy -1.3
  1. Find the remainder when \(x^3 + 2x\) is divided by \(x + 2\). [2]
  2. Write down the value of \(k\) for which \(x + 2\) is a factor of \(x^3 + 2x + k\). [1]