Factorisation After Division or Remainder

Questions that ask to factorise a polynomial completely, typically after using division results or adjusting by the remainder (e.g., factorise p(x) - r or p(x) + k).

4 questions · Moderate -0.1

1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02k Simplify rational expressions: factorising, cancelling, algebraic division
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CAIE P2 2020 March Q2
6 marks Moderate -0.8
2
  1. Find the quotient when \(4 x ^ { 3 } + 17 x ^ { 2 } + 9 x\) is divided by \(x ^ { 2 } + 5 x + 6\), and show that the remainder is 18 .
  2. Hence solve the equation \(4 x ^ { 3 } + 17 x ^ { 2 } + 9 x - 18 = 0\).
CAIE P2 2012 June Q3
5 marks Moderate -0.3
3
  1. Find the quotient when the polynomial $$8 x ^ { 3 } - 4 x ^ { 2 } - 18 x + 13$$ is divided by \(4 x ^ { 2 } + 4 x - 3\), and show that the remainder is 4 .
  2. Hence, or otherwise, factorise the polynomial $$8 x ^ { 3 } - 4 x ^ { 2 } - 18 x + 9$$
CAIE P2 2014 June Q3
6 marks Standard +0.3
3
  1. Find the quotient when \(6 x ^ { 4 } - x ^ { 3 } - 26 x ^ { 2 } + 4 x + 15\) is divided by ( \(x ^ { 2 } - 4\) ), and confirm that the remainder is 7 .
  2. Hence solve the equation \(6 x ^ { 4 } - x ^ { 3 } - 26 x ^ { 2 } + 4 x + 8 = 0\).
CAIE P2 2016 June Q4
7 marks Standard +0.3
4 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 8 x ^ { 3 } + 30 x ^ { 2 } + 13 x - 25$$
  1. Find the quotient when \(\mathrm { p } ( x )\) is divided by ( \(x + 2\) ), and show that the remainder is 5 .
  2. Hence factorise \(\mathrm { p } ( x ) - 5\) completely.
  3. Write down the roots of the equation \(\mathrm { p } ( | x | ) - 5 = 0\).