Notes & theory for this topic

Partial Fractions

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1. Decompose into partial fractions (distinct linear factors).
  1. \(y = \frac{7x - 10}{3x^{2} - 7x - 6}\)
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    Factorise the denominator: \(3x^{2} - 7x - 6 = (3x + 2)(x - 3)\). Write \(y = \frac{A}{3x + 2} + \frac{B}{x - 3}\), so \(7x - 10 = A(x - 3) + B(3x + 2)\). Matching coefficients gives \(A = 4,\; B = 1\).
    \(y = \frac{4}{3x + 2} + \frac{1}{x - 3}\)
  2. \(y = \frac{7x + 6}{6x^{2} + 15x + 9}\)
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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 \(7x + 6 = A(2x + 3) + B(3x + 3)\). Matching coefficients gives \(A = -1,\; B = 3\).
    \(y = -\frac{1}{3x + 3} + \frac{3}{2x + 3}\)
2. Decompose into partial fractions (a repeated factor).
  1. \(y = \frac{4x - 3}{4x^{2} - 4x + 1}\)
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    The denominator is \((2x - 1)^2\). Write \(y = \frac{A}{2x - 1} + \frac{B}{(2x - 1)^2}\), so \(4x - 3 = A(2x - 1) + B\). Matching coefficients gives \(A = 2,\; B = -1\).
    \(y = \frac{2}{2x - 1} - \frac{1}{(2x - 1)^{2}}\)
  2. \(y = \frac{2x + 4}{4x^{2} + 8x + 4}\)
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    The denominator is \((2x + 2)^2\). Write \(y = \frac{A}{2x + 2} + \frac{B}{(2x + 2)^2}\), so \(2x + 4 = A(2x + 2) + B\). Matching coefficients gives \(A = 1,\; B = 2\).
    \(y = \frac{1}{2x + 2} + \frac{2}{(2x + 2)^{2}}\)
3. Decompose into partial fractions (an irreducible quadratic factor).
  1. \(y = \frac{-3x^{2} + 6x + 12}{(x^{2} + 2x + 4)(2x + 2)}\)
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    The quadratic has no real roots, so write \(y = \frac{Ax+B}{x^{2} + 2x + 4} + \frac{C}{2x + 2}\). Then \(-3x^{2} + 6x + 12 = (Ax+B)(2x + 2) + C(x^{2} + 2x + 4)\). Matching coefficients gives \(A = -2,\; B = 4,\; C = 1\).
    \(y = \frac{-2x + 4}{x^{2} + 2x + 4} + \frac{1}{2x + 2}\)
  2. \(y = \frac{-6x^{2} - 5x - 4}{(x^{2} + 2)(x + 2)}\)
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    The quadratic has no real roots, so write \(y = \frac{Ax+B}{x^{2} + 2} + \frac{C}{x + 2}\). Then \(-6x^{2} - 5x - 4 = (Ax+B)(x + 2) + C(x^{2} + 2)\). Matching coefficients gives \(A = -3,\; B = 1,\; C = -3\).
    \(y = \frac{-3x + 1}{x^{2} + 2} - \frac{3}{x + 2}\)
4. Decompose into partial fractions (improper fraction — divide out first).
  1. \(y = \frac{8x^{2} - 9x - 7}{2x^{2} - x - 3}\)
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    The fraction is improper (top and bottom are both degree 2). Dividing out gives \(y = 4 + \frac{-5x + 5}{2x^{2} - x - 3}\). Factorise \(2x^{2} - x - 3 = (2x - 3)(x + 1)\) and decompose the remainder: \(A = -1,\; B = -2\).
    \(y = 4 - \frac{1}{2x - 3} - \frac{2}{x + 1}\)
  2. \(y = \frac{2x^{2} - x - 2}{2x^{2} + 3x + 1}\)
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    The fraction is improper (top and bottom are both degree 2). Dividing out gives \(y = 1 + \frac{-4x - 3}{2x^{2} + 3x + 1}\). Factorise \(2x^{2} + 3x + 1 = (2x + 1)(x + 1)\) and decompose the remainder: \(A = -2,\; B = -1\).
    \(y = 1 - \frac{2}{2x + 1} - \frac{1}{x + 1}\)
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