Notes & theory for this topic

Partial Fractions

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1. Decompose into partial fractions (distinct linear factors).
  1. \(y = \frac{-8x + 9}{3x^{2} + 7x - 6}\)
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    Factorise the denominator: \(3x^{2} + 7x - 6 = (x + 3)(3x - 2)\). Write \(y = \frac{A}{x + 3} + \frac{B}{3x - 2}\), so \(-8x + 9 = A(3x - 2) + B(x + 3)\). Matching coefficients gives \(A = -3,\; B = 1\).
    \(y = -\frac{3}{x + 3} + \frac{1}{3x - 2}\)
  2. \(y = \frac{-14x + 18}{6x^{2} - 15x + 9}\)
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    Factorise the denominator: \(6x^{2} - 15x + 9 = (3x - 3)(2x - 3)\). Write \(y = \frac{A}{3x - 3} + \frac{B}{2x - 3}\), so \(-14x + 18 = A(2x - 3) + B(3x - 3)\). Matching coefficients gives \(A = -4,\; B = -2\).
    \(y = -\frac{4}{3x - 3} - \frac{2}{2x - 3}\)
2. Decompose into partial fractions (a repeated factor).
  1. \(y = \frac{4x - 9}{x^{2} - 4x + 4}\)
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    The denominator is \((x - 2)^2\). Write \(y = \frac{A}{x - 2} + \frac{B}{(x - 2)^2}\), so \(4x - 9 = A(x - 2) + B\). Matching coefficients gives \(A = 4,\; B = -1\).
    \(y = \frac{4}{x - 2} - \frac{1}{(x - 2)^{2}}\)
  2. \(y = \frac{8x + 1}{4x^{2} + 4x + 1}\)
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    The denominator is \((2x + 1)^2\). Write \(y = \frac{A}{2x + 1} + \frac{B}{(2x + 1)^2}\), so \(8x + 1 = A(2x + 1) + B\). Matching coefficients gives \(A = 4,\; B = -3\).
    \(y = \frac{4}{2x + 1} - \frac{3}{(2x + 1)^{2}}\)
3. Decompose into partial fractions (an irreducible quadratic factor).
  1. \(y = \frac{2x^{2} + 6x + 12}{(x^{2} + x + 2)(x + 1)}\)
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    The quadratic has no real roots, so write \(y = \frac{Ax+B}{x^{2} + x + 2} + \frac{C}{x + 1}\). Then \(2x^{2} + 6x + 12 = (Ax+B)(x + 1) + C(x^{2} + x + 2)\). Matching coefficients gives \(A = -2,\; B = 4,\; C = 4\).
    \(y = \frac{-2x + 4}{x^{2} + x + 2} + \frac{4}{x + 1}\)
  2. \(y = \frac{-x^{2} + 8x - 19}{(x^{2} - x + 5)(x + 1)}\)
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    The quadratic has no real roots, so write \(y = \frac{Ax+B}{x^{2} - x + 5} + \frac{C}{x + 1}\). Then \(-x^{2} + 8x - 19 = (Ax+B)(x + 1) + C(x^{2} - x + 5)\). Matching coefficients gives \(A = 3,\; B = 1,\; C = -4\).
    \(y = \frac{3x + 1}{x^{2} - x + 5} - \frac{4}{x + 1}\)
4. Decompose into partial fractions (improper fraction — divide out first).
  1. \(y = \frac{12x^{2} + 24x + 17}{4x^{2} + 8x + 3}\)
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    The fraction is improper (top and bottom are both degree 2). Dividing out gives \(y = 3 + \frac{8}{4x^{2} + 8x + 3}\). Factorise \(4x^{2} + 8x + 3 = (2x + 3)(2x + 1)\) and decompose the remainder: \(A = -4,\; B = 4\).
    \(y = 3 - \frac{4}{2x + 3} + \frac{4}{2x + 1}\)
  2. \(y = \frac{4x^{2} - 8x - 4}{2x^{2} - 4x - 6}\)
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    The fraction is improper (top and bottom are both degree 2). Dividing out gives \(y = 2 + \frac{8}{2x^{2} - 4x - 6}\). Factorise \(2x^{2} - 4x - 6 = (x - 3)(2x + 2)\) and decompose the remainder: \(A = 1,\; B = -2\).
    \(y = 2 + \frac{1}{x - 3} - \frac{2}{2x + 2}\)
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