01
Distinct linear factors
Create one constant numerator for each distinct linear denominator factor.
Partial Fraction Decomposition Solver
Purpose-built solver page
Partial Fraction Decomposition Solver
Use this partial fraction decomposition solver to break a proper rational expression into simpler fractions with worked coefficient matching. It covers common linear, repeated, and irreducible quadratic factor patterns.
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Solve a problem nowpartial fraction decomposition solver
P(x)/Q(x) = Σ
Distinct linear factors
Repeated linear factors
Irreducible quadratic factors
partial fraction decomposition solver
Solve your partial fraction decomposition problem
Enter your partial fraction decomposition problem below. Add the variable, conditions, and method you want so the AI can return a more relevant worked response.
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Step
How to use the Partial Fraction Decomposition solver
A complete prompt usually produces a more useful explanation than an isolated expression.
- 1
Check the degrees
deg P < deg QIf the numerator degree is not smaller, divide first.
- 2
Factor the denominator
Q(x) = ∏ qᵢ(x)Factor completely over the requested number system.
- 3
Build the template
P/Q = Σ Nᵢ/qᵢInclude all terms required by distinct, repeated, and quadratic factors.
- 4
Solve and verify coefficients
Σ Nᵢ(Q/qᵢ) = PClear denominators, match coefficients or substitute convenient values, then recombine.
Tip: ask for a named method, a domain, or a verification step when your course requires one.
partial fraction decomposition solver
What this Partial Fraction Decomposition solver is designed for
The page is organized around the problem types people actually search for, while staying within the text-and-image solver's current capabilities.
Distinct linear factors
(x-a)(x-b) → A/(x-a) + B/(x-b)Create one constant numerator for each distinct linear denominator factor.
Repeated linear factors
(x-a)ⁿ → A₁/(x-a) + ⋯ + Aₙ/(x-a)ⁿInclude a term for every power from the first through the repeated factor's multiplicity.
Irreducible quadratic factors
x²+bx+c → (Ax+B)/(x²+bx+c)Use a linear numerator over each quadratic factor.
Improper rational expressions
deg P ≥ deg Q → P/Q = S + R/QPerform polynomial division before setting up the partial-fraction form.
Example
Partial Fraction Decomposition problems to try
These examples reflect common search intent and can be inserted into the solver as a starting point.
02
Repeated linear factors
Include a term for every power from the first through the repeated factor's multiplicity.
03
Irreducible quadratic factors
Use a linear numerator over each quadratic factor.
04
Improper rational expressions
Perform polynomial division before setting up the partial-fraction form.
Example 1
Decompose (3x + 5) / ((x - 1)(x + 2))A proper rational expression with two distinct linear factors.
Example 2
Decompose 1 / (x(x + 1)^2)A repeated linear factor requiring two separate powers.
Example 3
Decompose (2x^2 + 1) / ((x - 3)(x^2 + 4))A linear factor together with an irreducible quadratic factor.
Formula guides
Understand the formulas used here
Review the related formulas, when to use them, and how they connect in a worked example.
Quadratic Equation Formulas
Learn the essential quadratic equation formulas for solving, checking roots, and graphing a parabola, with one connected example.
Open formula guideExponent Rules
Learn the product, quotient, power, zero, negative, and fractional exponent rules with examples and one complete simplification.
Open formula guideCommon mistakes to watch for
A worked answer is most useful when you know which details are easy to lose along the way.
Skipping polynomial division
Partial-fraction setup begins with a proper rational expression.
Missing repeated powers
A repeated factor requires terms for every denominator power up to its multiplicity.
Using a constant over a quadratic
An irreducible quadratic factor needs a linear numerator such as Ax + B.
Partial Fraction Decomposition solver FAQ
