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Algebra : Partial Fractions

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The partial fractions command will rewrite a rational expression as a sum of terms with minimal denominators. A rational expression is a quotient of polynomials. Whenever the degree of the numerator of a rational expression is less than the degree of its denominator, the expression can be written as a sum of fractions whose denominators are the repeated linear or quadratic factors of the denominator of the original expression. If the degree of the numerator is greater than or equal to that of the denominator, polynomial long division is carried out first before partial fractions decomposition is attempted.

To use the partial fractions command, simply go to the basic partial fractions page, type in a rational expression and hit the "Partial Fractions" button. Your question will be automatically answered by computer and the reply will be shown in your browser within a few seconds. If you would like more control over how your question is answered, try the advanced partial fractions page, where there is an option which allows you to treat trigonometric functions as rational functions of exponentials in the partial fraction decomposition.

Examples

Here are some examples illustrating the types of rational expressions you can use the partial fractions command on and the results which QuickMath will return.

 Basic partial fractions command
Expression Result
x/(x-5)
      5
1 + ------
    -5 + x 
(x+5)/(x^2+x-2)
  2        1
------ - -----
-1 + x   2 + x 
1/(x^6-x^3)
    1         -3       -2 - x
---------- - x   + --------------
3 (-1 + x)                     2
                   3 (1 + x + x )
1/(x^2-a^2)
     1              1
------------ - -----------
2 a (-a + x)   2 a (a + x)
(x^2+2x-1)/(2x^3+3x^2-2x)
 1        1             1
--- - ---------- + ------------
2 x   10 (2 + x)   5 (-1 + 2 x)
1/(x^4-x^3)
  1       -3    -2   1
------ - x   - x   - -
-1 + x               x

 Advanced partial fractions command

Expression Option Result
1/(1-sin(2x)) Trig
                 -2
(cos(x) - sin(x))
1/(1-tan(2x)) Trig
         sin(2 x)
1 + -------------------
    cos(2 x) - sin(2 x)
1/(1-cos(4x)) Trig
      2         2
csc(x)    sec(x)
------- + -------
   8         8 

Options (advanced page only)

Trig functions

Values : checked or unchecked
Default : unchecked

When Trig functions is checked, the partial fractions command will treat trigonometric functions as rational functions of exponentials in the partial fraction decomposition. This allows expressions containing trigonometric functions to be decomposed into partial fractions when possible.