Partial Fraction Decomposition

About Partial Fraction Decomposition

Partial fraction decomposition breaks a complex rational function into simpler fractions that are easier to integrate.

How to use this calculator

1. Enter the rational function you want to decompose.
2. Factor the denominator completely if it is not already factored.
3. Check that the fraction is proper, meaning the degree of the numerator is less than the degree of the denominator.
4. Read the decomposed form and use it to simplify your work.

The formula explained

For a fraction like \(\frac{P(x)}{(x-a)(x-b)}\), the goal is to write it as \(\frac{A}{x-a}+\frac{B}{x-b}\). The constants \(A\) and \(B\) are chosen so the two sides are equal for every allowed value of \(x\).

• x = the variable in the rational function
• P(x) = the numerator polynomial
• a = a value that makes one denominator factor zero
• b = another value that makes one denominator factor zero
• A = the constant coefficient of the first partial fraction
• B = the constant coefficient of the second partial fraction

Step by step method

1. Factor the denominator completely, for example into terms like \((x-a)(x-b)\).
2. Set up the partial fraction form with unknown constants, such as \(\frac{A}{x-a}+\frac{B}{x-b}\).
3. Multiply both sides by the full denominator to clear fractions, then solve for the constants using substitution or matching coefficients.

Worked example

Problem. Decompose \(\frac{5x+1}{(x-2)(x+3)}\) into partial fractions.

1. Write \(\frac{5x+1}{(x-2)(x+3)}=\frac{A}{x-2}+\frac{B}{x+3}\).
2. Multiply by \((x-2)(x+3)\): \(5x+1=A(x+3)+B(x-2)\).
3. Choose values that simplify the equation. If \(x=2\), then \(11=5A\), so \(A=\frac{11}{5}\). If \(x=-3\), then \(-14=-5B\), so \(B=\frac{14}{5}\).

Tips and common mistakes

• If the numerator degree is at least the denominator degree, divide first before decomposing.
• Always factor the denominator fully, because partial fractions depend on the individual factors.

Frequently asked questions