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Partial Fraction Decomposition

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A rational function P(x)/Q(x) can be rewritten using what is known as partial fraction decomposition. This procedure often allows integration to be performed on each term separately by inspection. For each factor of Q(x) the form (ax+b)^m, introduce terms

(A_1)/(ax+b)+(A_2)/((ax+b)^2)+...+(A_m)/((ax+b)^m).
(1)

For each factor of the form (ax^2+bx+c)^m, introduce terms

(A_1x+B_1)/(ax^2+bx+c)+(A_2x+B_2)/((ax^2+bx+c)^2)+...+(A_mx+B_m)/((ax^2+bx+c)^m).
(2)

Then write

(P(x))/(Q(x))=(A_1)/(ax+b)+...+(A_2x+B_2)/(ax^2+bx+c)+...
(3)

and solve for the A_is and B_is.

Partial fraction decomposition is implemented in the Wolfram Language as Apart.

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References

Blake, S. "Step-by-Step Partial Fractions." http://calc101.com/webMathematica/partial-fractions.jsp. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 13-15, 1987.

Referenced on Wolfram|Alpha

Partial Fraction Decomposition

Cite this as:

Weisstein, Eric W. "Partial Fraction Decomposition." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PartialFractionDecomposition.html

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