Integration or area using factorised polynomial

A question is this type if and only if you must use a factorised polynomial to find an area or evaluate an integral, after first factorising.

3 questions

OCR C2 2015 June Q6
6 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = x ^ { 3 } - 19 x + 30\).
  1. Given that \(x = 2\) is a root of the equation \(\mathrm { f } ( x ) = 0\), express \(\mathrm { f } ( x )\) as the product of 3 linear factors.
  2. Use integration to find the exact value of \(\int _ { - 5 } ^ { 3 } \mathrm { f } ( x ) \mathrm { d } x\).
  3. Explain with the aid of a sketch why the answer to part (ii) does not give the area enclosed by the curve \(y = \mathrm { f } ( x )\) and the \(x\)-axis for \(- 5 \leqslant x \leqslant 3\).
Edexcel AS Paper 1 2020 June Q10
10. $$g ( x ) = 2 x ^ { 3 } + x ^ { 2 } - 41 x - 70$$
  1. Use the factor theorem to show that \(\mathrm { g } ( x )\) is divisible by \(( x - 5 )\).
  2. Hence, showing all your working, write \(\mathrm { g } ( x )\) as a product of three linear factors. The finite region \(R\) is bounded by the curve with equation \(y = \mathrm { g } ( x )\) and the \(x\)-axis, and lies below the \(x\)-axis.
  3. Find, using algebraic integration, the exact value of the area of \(R\).
SPS SPS SM Pure 2021 June Q10
10. $$g ( x ) = 2 x ^ { 3 } + x ^ { 2 } - 41 x - 70$$
  1. Use the factor theorem to show that \(\mathrm { g } ( x )\) is divisible by \(( x - 5 )\).
  2. Hence, showing all your working, write \(\mathrm { g } ( x )\) as a product of three linear factors. The finite region \(R\) is bounded by the curve with equation \(y = \mathrm { g } ( x )\) and the \(x\)-axis, and lies below the \(x\)-axis.
  3. Find, using algebraic integration, the exact value of the area of \(R\).
    [0pt] [BLANK PAGE]