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).

2 questions · Standard +0.0

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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 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 \(\mathrm { p } ( | x | ) - 5 = 0\).