Partial Fraction Decomposition Calculator
Use this Partial Fraction Decomposition Calculator to enter values, adjust options, and review results in a compact responsive workspace.
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Result
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What Is Partial Fraction Decomposition?
Partial fraction decomposition is a method for rewriting a rational function as a sum of simpler rational expressions. A rational function is a fraction whose numerator and denominator are polynomials, such as:
$$
\frac{5x + 3}{x^2 + 5x + 6}
$$
The goal is not to change the value of the expression. The goal is to express the same rational function in a form that is easier to work with. For example, a denominator that factors into smaller pieces can often be rewritten as separate fractions with those smaller pieces in the denominators.
Partial fractions are especially useful in calculus because many rational functions are difficult to integrate in their original form but become much easier after decomposition. They are also useful in algebra, differential equations, engineering, control systems, and any topic where rational functions need to be simplified or analyzed.
A partial fraction decomposition is normally used after the denominator has been factored. The factor pattern determines what kinds of simpler fractions are needed.
Why Partial Fraction Decomposition Matters
Partial fraction decomposition turns one complicated rational expression into several simpler ones. That makes it easier to:
- integrate rational functions in calculus;
- compare algebraic forms of the same expression;
- work with repeated factors in denominators;
- separate a rational function into terms with simpler denominators;
- check algebraic simplifications more clearly.
For students, the method is also a practical way to connect several algebra skills at once: polynomial division, factoring, systems of linear equations, and coefficient matching.
Key Terms to Know
- Polynomial: An expression made from constants and whole-number powers of a variable, such as \(x^2 + 5x + 6\).
- Rational function: A function that can be written as \(\frac{N(x)}{D(x)}\), where \(N(x)\) and \(D(x)\) are polynomials and \(D(x) \ne 0\).
- Proper rational function: A rational function where the degree of the numerator is less than the degree of the denominator.
- Improper rational function: A rational function where the degree of the numerator is greater than or equal to the degree of the denominator.
- Polynomial long division: The process used to rewrite an improper rational function as a polynomial plus a proper rational function.
- Linear factor: A first-degree factor, such as \(x - 2\) or \(x + 5\).
- Repeated linear factor: A linear factor raised to a power, such as \((x - 2)^3\).
- Irreducible quadratic factor: A quadratic factor that cannot be factored into real linear factors, such as \(x^2 + 1\).
- Coefficient matching: A method where both sides of a polynomial identity are compared by matching the coefficients of like powers of \(x\).
How Partial Fraction Decomposition Works
Start with a rational function:
$$
\frac{N(x)}{D(x)}
$$
where \(N(x)\) is the numerator polynomial and \(D(x)\) is the denominator polynomial.
If the rational function is improper, divide first:
$$
\frac{N(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)}
$$
where:
- \(Q(x)\) is the polynomial quotient;
- \(R(x)\) is the remainder;
- \(\deg(R) < \deg(D)\).
Only the proper fraction \(\frac{R(x)}{D(x)}\) is decomposed into partial fractions.
Next, factor the denominator. The denominator factor pattern determines the partial fraction form.
For a repeated linear factor \((x - a)^m\), include a term for every power from \(1\) to \(m\):
$$
\frac{A_1}{x-a} + \frac{A_2}{(x-a)^2} + \cdots + \frac{A_m}{(x-a)^m}
$$
For an irreducible quadratic factor such as \(q(x) = ax^2 + bx + c\), the numerator must be linear:
$$
\frac{Bx + C}{q(x)}
$$
The unknown letters, such as \(A_1\), \(B\), and \(C\), are coefficients. To find them, multiply both sides of the decomposition by the full denominator. This clears the fractions and creates a polynomial identity. Then match coefficients of the same powers of \(x\) and solve the resulting system of equations.
Examples of Partial Fraction Decomposition in Practice
Example 1: A Simple Proper Rational Function
Decompose:
$$
\frac{5x + 3}{x^2 + 5x + 6}
$$
First factor the denominator:
$$
x^2 + 5x + 6 = (x + 2)(x + 3)
$$
Because the denominator has two distinct linear factors, use this form:
$$
\frac{5x + 3}{(x + 2)(x + 3)} = \frac{A}{x + 2} + \frac{B}{x + 3}
$$
Multiply both sides by \((x + 2)(x + 3)\):
$$
5x + 3 = A(x + 3) + B(x + 2)
$$
Expand and match coefficients:
$$
5x + 3 = (A + B)x + (3A + 2B)
$$
So:
$$
A + B = 5
$$
$$
3A + 2B = 3
$$
Solving gives \(A = -7\) and \(B = 12\). Therefore:
$$
\frac{5x + 3}{x^2 + 5x + 6} = -\frac{7}{x + 2} + \frac{12}{x + 3}
$$
Example 2: An Improper Rational Function
Decompose:
$$
\frac{x^3 + 2x^2 + 4x + 3}{x^2 + 3x + 2}
$$
The numerator has degree \(3\), and the denominator has degree \(2\), so the rational function is improper. Divide first:
$$
\frac{x^3 + 2x^2 + 4x + 3}{x^2 + 3x + 2} = x - 1 + \frac{5x + 5}{x^2 + 3x + 2}
$$
Now factor the denominator:
$$
x^2 + 3x + 2 = (x + 1)(x + 2)
$$
Set up the proper fractional part:
$$
\frac{5x + 5}{(x + 1)(x + 2)} = \frac{A}{x + 1} + \frac{B}{x + 2}
$$
Multiplying through gives:
$$
5x + 5 = A(x + 2) + B(x + 1)
$$
Matching coefficients leads to \(A = 0\) and \(B = 5\). The decomposition is:
$$
\frac{x^3 + 2x^2 + 4x + 3}{x^2 + 3x + 2} = x - 1 + \frac{5}{x + 2}
$$
The polynomial term \(x - 1\) appears because the original rational function was improper.
Example 3: A Quadratic Factor
Decompose:
$$
\frac{2x^2 + 7x + 6}{x^3 + x}
$$
Factor the denominator:
$$
x^3 + x = x(x^2 + 1)
$$
The factor \(x\) is linear, and \(x^2 + 1\) is an irreducible quadratic over the real numbers. Use a constant numerator over the linear factor and a linear numerator over the quadratic factor:
$$
\frac{2x^2 + 7x + 6}{x(x^2 + 1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1}
$$
Multiply by \(x(x^2 + 1)\):
$$
2x^2 + 7x + 6 = A(x^2 + 1) + (Bx + C)x
$$
Expand:
$$
2x^2 + 7x + 6 = (A + B)x^2 + Cx + A
$$
Matching coefficients gives \(A = 6\), \(C = 7\), and \(B = -4\). So:
$$
\frac{2x^2 + 7x + 6}{x^3 + x} = \frac{6}{x} + \frac{-4x + 7}{x^2 + 1}
$$
How to Interpret the Result
A partial fraction result is an algebraically equivalent form of the original rational function, except at values of \(x\) where the original denominator is zero. Those excluded values still matter because rewriting the expression does not remove the original domain restrictions.
If the result begins with a polynomial term, the original rational function was improper and polynomial division was used first. For example, a result like:
$$
x - 1 + \frac{5}{x + 2}
$$
means the rational function has a polynomial part, \(x - 1\), plus a simpler fractional part.
If the result contains several fractions, each fraction corresponds to a factor or power of a factor in the denominator. Repeated factors usually create multiple terms, such as:
$$
\frac{A}{x-a} + \frac{B}{(x-a)^2}
$$
A term with a linear numerator over a quadratic denominator, such as:
$$
\frac{Bx + C}{x^2 + 1}
$$
means the denominator has a quadratic factor that is being kept as a quadratic rather than split into real linear factors.
When coefficients are shown as decimals, treat them as rounded display values unless the result clearly shows exact integers. Very small differences may be caused by numerical solving and display rounding.
Common Mistakes and Misconceptions
One common mistake is trying to decompose an improper rational function before doing polynomial division. Partial fraction decomposition is applied to a proper rational function, so long division comes first when the numerator degree is too high.
Another mistake is using the wrong form for the denominator factors. A repeated factor such as \((x + 1)^2\) does not get only one term. It needs both:
$$
\frac{A}{x + 1} + \frac{B}{(x + 1)^2}
$$
A quadratic factor also needs a numerator with \(x\) in it. For example, over \(x^2 + 1\), use:
$$
\frac{Bx + C}{x^2 + 1}
$$
not just:
$$
\frac{A}{x^2 + 1}
$$
It is also easy to forget that a decomposition is an identity in \(x\). After clearing denominators, the coefficients of matching powers of \(x\) must agree on both sides.
Finally, do not assume that the number of factor groups is always the same as the number of final partial fraction terms. One repeated factor group can create several terms.
When to Use Partial Fraction Decomposition
Use partial fraction decomposition when you need to rewrite a rational function into simpler rational pieces.
It is especially useful when:
- integrating rational functions in calculus;
- simplifying expressions with factored polynomial denominators;
- studying rational functions with repeated linear factors;
- solving problems that require coefficient matching;
- preparing rational expressions for algebraic comparison;
- working with differential equations or transforms that involve rational expressions.
It is usually not the first tool to use if the expression can be simplified by canceling a common factor, factoring a numerator, or using a simpler algebraic identity.
Limitations and Things to Keep in Mind
Partial fraction decomposition depends on the factorization of the denominator. If the denominator cannot be factored into a supported form, the decomposition may be difficult or unavailable through an automatic tool.
This calculator is designed for polynomial expressions in the single variable \(x\). It expects the numerator and denominator to be entered separately. It does not accept one combined fraction such as \(\frac{1}{x+1}\) as a polynomial input.
The calculator supports expanded polynomial degree at most \(12\) and powers from \(0\) through \(12\). It accepts \(x\), decimals, scientific notation, multiplication, parentheses, and division by nonzero numeric constants such as \(x/2\). It does not support other variables, trigonometric functions, logarithms, radicals, negative powers, fractional powers, or division by a polynomial.
Denominators are decomposed over the real numbers into linear and irreducible quadratic factors. Multiple and repeated quadratic factors are supported. Exact rational factors use exact arithmetic; irrational factors use verified decimal approximations. If numerical factorization or reconstruction cannot be verified, the calculator reports an error instead of displaying a tentative result.
Approximate irrational coefficients are rounded to a readable number of significant digits. The result status identifies exact and approximate decompositions, and the verification step reports reconstruction error. Original real denominator zeros remain excluded even when a common factor cancels.
How to Use This Calculator
- Enter the numerator polynomial in the numerator field using \(x\) as the variable.
- Enter the denominator polynomial in the denominator field using \(x\) as the variable.
- Use polynomial notation such as \(+\), \(-\), \(*\), parentheses, implicit multiplication, and powers like \(x^2\).
- Choose an example case if you want the fields filled automatically.
- Review the decomposition result, original rational function, proper numerator after division, factor status, coefficient table, and calculation steps.
- If an error appears, check the syntax, make sure the denominator contains \(x\), and use a denominator that factors into supported linear and quadratic parts.
Frequently Asked Questions
What is the difference between a proper and improper rational function?
A proper rational function has a numerator degree smaller than the denominator degree. An improper rational function has a numerator degree greater than or equal to the denominator degree. Improper rational functions need polynomial division before the partial fraction part is set up.
Why do repeated factors need more than one partial fraction term?
A repeated factor represents several powers of the same denominator factor. For \((x-a)^m\), the decomposition needs terms with denominators \((x-a)\), \((x-a)^2\), and so on through \((x-a)^m\). Leaving out one of those terms can make the coefficient system impossible or incorrect.
Why does a quadratic factor use a numerator like \(Bx + C\)?
The numerator over a quadratic factor must have degree less than the quadratic denominator. The most general polynomial with degree less than \(2\) is linear, so the numerator is written as \(Bx + C\). Using only a constant numerator may miss valid decompositions.
Does partial fraction decomposition change the domain?
No. The decomposed form is equivalent to the original rational function only where the original denominator is not zero. Values that make the original denominator zero remain excluded, even if a simplified-looking term appears later.
Why did I get a polynomial term before the fractions?
A polynomial term appears when the original rational function is improper. The numerator was divided by the denominator first, producing a quotient plus a proper rational remainder. The partial fraction decomposition is then applied to the remainder fraction.
Why are some coefficients displayed as decimals?
Some coefficients may be solved numerically and displayed in decimal form for readability. Integer values are shown as integers, while non-integer values may be rounded. If you need exact fractions, verify the result algebraically or use a symbolic method that preserves rational coefficients.
Sources and References
Books
- Gilbert Strang and Edwin “Jed” Herman. Calculus Volume 2. OpenStax, 2016. Section 3.4, “Partial Fractions.” https://openstax.org/books/calculus-volume-2/pages/3-4-partial-fractions
- Gregory Hartman, Sean Fitzpatrick, Alex Jordan, and Carly Vollet. APEX Calculus. Section 6.5, “Partial Fraction Decomposition.” https://opentext.uleth.ca/apex-video/secpartialfraction.html
Online and Official Sources
- The University of Texas at Austin, Mathematics Department. “Partial Fractions.” M408S Current Course Notes, accessed July 4, 2026. https://web.ma.utexas.edu/users/m408s/CurrentWeb/LM7-4-2.php