Partial fractions with quadratic factor

A question is this type if and only if the denominator contains an irreducible quadratic factor requiring a partial fraction of the form (Ax + B)/(quadratic).

3 questions · Standard +0.3

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CAIE P3 2006 June Q9
10 marks Standard +0.3
  1. Express \(\frac{10}{(2-x)(1+x^2)}\) in partial fractions. [5]
  2. Hence, given that \(|x| < 1\), obtain the expansion of \(\frac{10}{(2-x)(1+x^2)}\) in ascending powers of \(x\), up to and including the term in \(x^3\), simplifying the coefficients. [5]
CAIE P3 2017 June Q8
10 marks Standard +0.3
Let \(\mathrm{f}(x) = \frac{5x^2 - 7x + 4}{(3x + 2)(x^2 + 5)}\).
  1. Express \(\mathrm{f}(x)\) in partial fractions. [5]
  2. Hence obtain the expansion of \(\mathrm{f}(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\). [5]
CAIE P3 2013 November Q7
10 marks Standard +0.3
Let \(f(x) = \frac{2x^2 - 7x - 1}{(x-2)(x^2+3)}\).
  1. Express \(f(x)\) in partial fractions. [5]
  2. Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\). [5]