How to use the Partial Fraction Decomposition Calculator
- 1. Enter the fractionType the numerator and the denominator in x (or s). The denominator can be factored or expanded.
- 2. Press DecomposeThe tool factors the denominator, writes the form with A, B, C… and compares coefficients.
- 3. Read the stepsCopy the form, the equations, the constants and the final answer into your notebook.
Rules for the form of partial fractions
First make sure the fraction is proper: the numerator's degree must be lower than the denominator's. If not, divide and keep the quotient. Then factor the denominator and write one term for each factor:
- Linear factor (x − a): A/(x − a).
- Repeated factor (x − a)³: A/(x − a) + B/(x − a)² + C/(x − a)³.
- Irreducible quadratic (x² + px + q): (Bx + C)/(x² + px + q).
- Repeated quadratic (x² + px + q)²: (Bx + C)/(x² + px + q) + (Dx + E)/(x² + px + q)².
Multiply both sides by the denominator and compare the coefficients of each power of x. That gives as many equations as unknowns.
Worked example
Decompose (3x + 5)/((x − 1)(x + 2)²). The form is A/(x − 1) + B/(x + 2) + C/(x + 2)². Multiplying out: 3x + 5 = A(x + 2)² + B(x − 1)(x + 2) + C(x − 1).
Comparing coefficients: x²: A + B = 0; x: 4A + B + C = 3; constant: 4A − 2B − C = 5. Putting x = 1 gives 8 = 9A, so A = 8/9; putting x = −2 gives −1 = −3C, so C = 1/3; then B = −8/9. The answer is (8/9)/(x − 1) − (8/9)/(x + 2) + (1/3)/(x + 2)².
Where partial fractions are used
They turn hard integrals into log and arctan terms, and they are the first step of the inverse Laplace transform and the inverse Z-transform in signals and control. For fraction arithmetic with numbers, use the fraction calculator.
Frequently asked questions
Can I enter the denominator expanded?
Yes. Quadratics are factored with the formula, and integer cubics or quartics are factored with the rational root test. Other high-degree factors are split numerically.
What about factors like 3x − 1?
They work. The working uses (x − 1/3), and the final answer is written back with the integer factor, for example (3/5)/(3x − 1).
What if the degree of the numerator is too high?
The tool does polynomial long division first and shows the quotient and remainder, then splits the remainder.
Can I use s instead of x?
Yes. If the input uses s, the whole working is done in s, which is handy for Laplace problems.
Is this calculator free?
Yes. You fill a short form once when you see your first result, and other Uncodemy tools then open without the form. Everything is calculated in your browser.