Improper fraction partial fractions

A question is this type if and only if the rational function is improper (numerator degree ≥ denominator degree) requiring the form A + B/(x-p) + C/(x-q) in the partial fraction decomposition.

4 questions · Standard +0.3

1.02y Partial fractions: decompose rational functions
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Edexcel C34 2019 January Q2
10 marks Standard +0.3
2. Given that $$\frac { 3 x ^ { 2 } + 4 x - 7 } { ( x + 1 ) ( x - 3 ) } \equiv A + \frac { B } { x + 1 } + \frac { C } { x - 3 }$$
  1. find the values of the constants \(A , B\) and \(C\).
  2. Hence, or otherwise, find the series expansion of $$\frac { 3 x ^ { 2 } + 4 x - 7 } { ( x + 1 ) ( x - 3 ) } \quad | x | < 1$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\) Give each coefficient as a simplified fraction.
Edexcel C4 2010 June Q5
11 marks Standard +0.3
5. $$\frac { 2 x ^ { 2 } + 5 x - 10 } { ( x - 1 ) ( x + 2 ) } \equiv A + \frac { B } { x - 1 } + \frac { C } { x + 2 }$$
  1. Find the values of the constants \(A , B\) and \(C\).
  2. Hence, or otherwise, expand \(\frac { 2 x ^ { 2 } + 5 x - 10 } { ( x - 1 ) ( x + 2 ) }\) in ascending powers of \(x\), as far as the term in \(x ^ { 2 }\). Give each coefficient as a simplified fraction.
CAIE P3 2017 November Q8
9 marks Standard +0.3
Let \(\text{f}(x) = \frac{4x^2 + 9x - 8}{(x + 2)(2x - 1)}\).
  1. Express \(\text{f}(x)\) in the form \(A + \frac{B}{x + 2} + \frac{C}{2x - 1}\). [4]
  2. Hence show that \(\int_1^4 \text{f}(x) \, dx = 6 + \frac{1}{2} \ln\left(\frac{16}{7}\right)\). [5]
Edexcel C4 Q8
14 marks Standard +0.3
$$\text{f}(x) = \frac{x(3x-7)}{(1-x)(1-3x)}, \quad |x| < \frac{1}{3}.$$
  1. Find the values of the constants \(A\), \(B\) and \(C\) such that $$\text{f}(x) = A + \frac{B}{1-x} + \frac{C}{1-3x}.$$ [4]
  2. Evaluate $$\int_0^{\frac{1}{4}} \text{f}(x) \, dx,$$ giving your answer in the form \(p + \ln q\), where \(p\) and \(q\) are rational. [5]
  3. Find the series expansion of f(x) in ascending powers of \(x\) up to and including the term in \(x^3\), simplifying each coefficient. [5]