Partial fraction decomposition

Partial fraction decomposition

Lessons

\bullet Partial fraction decomposition expresses a rational function f(x)g(x)\frac{f(x)}{g(x)}, where f(x)f(x) and g(x)g(x) are polynomials in xx, as a sum of simpler fractions.

\bullet Partial fraction decomposition only applies to proper fractions in which the degree of the numerator is less than that of the denominator.

  • 1.
    Introduction to Partial Fraction Decomposition
    a)
    What is partial fraction decomposition?

    b)
    When can we perform partial fraction decomposition?


  • 2.
    Case 1: Denominator is a product of linear factors with no repeats

    Find the partial fractions of:

    a)
    x+7(x+3)(x1)\frac{x + 7}{(x + 3)(x - 1)}

    b)
    4x+3x2+x\frac{4x + 3}{x^{2} + x}


  • 3.
    Case 2: Denominator is a product of linear factors with repeats

    Find the partial fractions of :

    a)
    3x25(x2)3\frac{3x^{2} - 5}{(x - 2)^{3}}

    b)
    2x1x2+10x+25\frac{2x - 1}{x^{2} + 10x + 25}


  • 4.
    Case 3: Denominator contains irreducible quadratic factors with no repeats

    Find the partial fractions of :

    2x2+5x+8x38x\frac{2x^{2} + 5x + 8}{x^{3} - 8x}


  • 5.
    Case 4: Denominator contains irreducible quadratic factors with repeats

    Find the partial fractions of:

    3x4+x3+1x(x2+1)2\frac{3x^{4} + x^{3} + 1}{x(x^{2} + 1)^{2}}


  • 6.
    First perform long division, then partial fraction decomposition

    Find the partial fractions of:

    a)
    x33x2+4xx23x2\frac{x^{3} - 3x^{2} + 4x}{x^{2} - 3x 2}

    b)
    2x2+14x+24x2+6x16\frac{2x^{2} + 14x + 24}{x^{2} + 6x - 16}