Improper fraction with linear factors – division then partial fractions

Numerator degree equals denominator degree or exceeds it by one; perform polynomial division to extract a polynomial part, then decompose the remainder over linear factors into partial fractions.

5 questions · Moderate -0.1

1.02y Partial fractions: decompose rational functions
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CAIE P3 2024 June Q2
5 marks Standard +0.3
2 Express \(\frac { 6 x ^ { 2 } - 9 x - 16 } { 2 x ^ { 2 } - 5 x - 12 }\) in partial fractions.
CAIE P3 2021 November Q4
5 marks Moderate -0.5
4 Express \(\frac { 4 x ^ { 2 } - 13 x + 13 } { ( 2 x - 1 ) ( x - 3 ) }\) in partial fractions.
Edexcel C4 2013 January Q3
4 marks Standard +0.3
3. Express \(\frac { 9 x ^ { 2 } + 20 x - 10 } { ( x + 2 ) ( 3 x - 1 ) }\) in partial fractions.
OCR C4 2015 June Q10
14 marks Standard +0.3
10
  1. Express \(\frac { x + 8 } { x ( x + 2 ) }\) in partial fractions.
  2. By first using division, express \(\frac { 7 x ^ { 2 } + 16 x + 16 } { x ( x + 2 ) }\) in the form \(P + \frac { Q } { x } + \frac { R } { x + 2 }\). A curve has parametric equations \(x = \frac { 2 t } { 1 - t } , y = 3 t + \frac { 4 } { t }\).
  3. Show that the cartesian equation of the curve is \(y = \frac { 7 x ^ { 2 } + 16 x + 16 } { x ( x + 2 ) }\).
  4. Find the area of the region bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 2\). Give your answer in the form \(L + M \ln 2 + N \ln 3\).
CAIE P3 2024 June Q5
5 marks Moderate -0.8
Express \(\frac{6x^2 - 2x + 2}{(x - 1)(2x + 1)}\) in partial fractions. [5]