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Enter your values, then click Calculate result.How this calculator helps
Decompose a rational polynomial over supplied real linear factors, including repeated roots and an improper polynomial quotient. Enter the numerator’s coefficients in descending powers, including zeros for missing terms. For 2x²+3, enter 2, 0, 3. Enter denominator roots as a separate list: roots 1 and minus 1 mean the monic denominator (x−1)(x+1). The page does not accept a typed algebraic expression in the coefficient field or factor an arbitrary denominator automatically. If the denominator’s leading coefficient is not one, divide the numerator by that coefficient before using its roots.
How to use it
- 1
Enter numerator coefficients from highest power to constant, including zeros for missing powers.
- 2
Enter denominator real roots separated by commas. Repeat a root for its multiplicity; the denominator is monic.
- 3
Click Calculate result to submit the inputs. Editing fields keeps the last submitted output until you click again.
- 4
Check the worked example, units and method limits before using the result.
Formula and methodology
Build Q(x)=∏(x−root). Divide numerator P by Q first when needed. Solve P_remainder=Σ A(r,k) × Q(x)/(x−r)^k by matching coefficients.
The calculator applies the disclosed heuristic to values entered on this device. It does not load private admissions data or claim to reproduce an institution's review process.
Worked calculation example
Numerator coefficients 1 and roots 1, −1 represent 1/(x²−1), decomposed as 0.5/(x−1) − 0.5/(x+1), excluding x=±1.
How to interpret your result
Monic denominator with 1–6 supplied real linear roots only; numerator up to degree eight. No automatic factoring or irreducible quadratic factors. Displayed numerical coefficients are rounded. The examples identify the method and input basis, so the result can be checked without guessing a hidden convention.
For different inputs or formulas, use Fraction Calculator; Decimals Calculator; Implicit Differentiation Calculator.
Related questions this calculator covers
- partial fraction calculator
Scenario comparison
| Scenario | What it shows |
|---|---|
| 1/(x²−1) | 0.5/(x−1) − 0.5/(x+1). |
| 1/(x−1)² | first-power coefficient zero, second-power coefficient one. |
| x²/(x−1) | polynomial quotient x+1 and remainder 1/(x−1). |
Common mistakes to avoid
- Leaving out zero coefficients for missing powers.
- Entering factor constants instead of roots.
- Dropping the polynomial quotient or original excluded values.
Substitute the same inputs into the disclosed equation and compare with this example: Numerator coefficients 1 and roots 1, −1 represent 1/(x²−1), decomposed as 0.5/(x−1) − 0.5/(x+1), excluding x=±1. Keep the method, units and source assumptions with any recorded result.
Authoritative reference. Monic denominator with 1–6 supplied real linear roots only; numerator up to degree eight. No automatic factoring or irreducible quadratic factors. Displayed numerical coefficients are rounded.What can affect the result?
Simple and repeated linear factors
A simple root contributes a term A/(x−r). A root repeated twice needs terms for both the first and second denominator powers. Repeat the same root in the input list to specify multiplicity. The calculator builds a coefficient system using those terms and solves for the displayed constants. It supports up to six real roots counting repetitions. Irreducible quadratic factors need a different numerator structure and are outside this implementation’s stated scope.
Improper rational functions
When the numerator degree is at least the denominator degree, polynomial division comes first. The result contains a polynomial quotient plus a proper remainder fraction, and only that remainder is decomposed into partial fractions. Keeping the quotient prevents a common error where the fractional terms describe only part of the original function. The output displays the expanded denominator and quotient so you can verify the representation before using it in another calculation.
Domain and numerical verification
Every root of the original denominator remains excluded, even if algebraic cancellation makes an expression look defined there. Coefficients are calculated numerically and rounded for display, so the result is an approximate printed decomposition rather than an exact symbolic rational proof. Closely spaced or extreme roots can be ill-conditioned and are rejected when the linear system is unreliable. Verify by recombining the fractions or evaluating both forms at non-root values; do not test at excluded points.
Privacy and browser processing
Values entered on this page are processed in the current browser session. SolvePilot does not require an account and does not receive the values entered into the calculator. Refreshing or closing the page clears the working values unless the browser itself restores a previous session. Avoid entering identifying or account information because the calculation needs summary values only.
Accuracy and verification
Accuracy depends first on input quality. Confirm definitions, scales, dates and source information before entering a value. Keep an independent record of any result used for planning because this page does not create an official statement or retain a calculation history.
Limits of this estimate
Monic denominator with 1–6 supplied real linear roots only; numerator up to degree eight. No automatic factoring or irreducible quadratic factors. Displayed numerical coefficients are rounded.
Sources and review information
Frequently asked questions
Can I paste a fraction like 1/(x²−1)?＋
This version uses explicit coefficient and root lists. Enter 1 for the numerator and 1, −1 for the denominator roots. The input format avoids guessing a denominator factorization and keeps the supported real-linear-factor scope clear.
How do I enter a repeated factor?＋
Repeat the root. For (x−1)²(x+2), enter roots 1, 1, −2. The output includes terms with (x−1) and (x−1)², plus the factor at −2. Omitting a repeated root would represent a different denominator.
What if numerator degree is higher?＋
The calculator performs polynomial division first and displays the quotient. The remaining proper fraction is then decomposed. Do not discard the quotient when checking the original rational function or using the decomposition in a later algebra step.
Does it support irreducible quadratics?＋
No. This bounded implementation accepts supplied real linear roots only. A factor such as x²+1 has no real roots and requires a linear numerator over the quadratic factor. The page does not fabricate a real-root factorization for that case.
Why are the results decimals?＋
The coefficient system is solved numerically and displayed to a limited number of decimal places. Simple examples often have terminating constants, but other inputs can require rounded coefficients. Recombination checks should allow for displayed rounding and avoid poorly conditioned root sets.