How to use
Enter the numerator coefficients from highest power to constant, separated by commas. For 3x + 5, enter 3, 5. Then enter the integer roots of the denominator factors. Roots 1, -2, and 4 represent (x − 1)(x + 2)(x − 4). The number of numerator coefficients must not exceed the number of roots, which keeps the rational fraction proper.
The gray values initially shown are a working example, not saved input. Focusing either input clears both examples so your own values are visually distinct. Calculate validates the complete fraction and confirms a copyable result. Editing either field immediately removes the prior result and disables Copy. Clear empties the inputs and result; it does not silently restore the example.
Method and result reading
For distinct linear factors, the decomposition has one term Aᵢ/(x − rᵢ) per root. The calculator uses the cover-up identity Aᵢ = N(rᵢ) / ∏ⱼ≠ᵢ(rᵢ − rⱼ). Every numerator and denominator is stored as a BigInt and reduced by the greatest common divisor, so values such as 8/3 remain exact rather than becoming 2.67.
The main result shows the full sum. Exact coefficients lists A₁, A₂, and so on in the same order as the entered roots. Recombined numerator gives the recovered coefficient list after putting the terms over a common denominator. Identity check compares that list with the original numerator, including leading zero coefficients needed for the denominator degree.
Worked examples
For numerator 3, 5 and roots 1, -2, the input is (3x + 5)/[(x − 1)(x + 2)]. Substitution gives A = 8/3 and B = 1/3, so the result is 8/3 divided by (x − 1), plus 1/3 divided by (x + 2). Recombining gives 3x + 5 exactly.
For numerator 1, 0, 0 and roots 1, 2, 3, the fraction x²/[(x − 1)(x − 2)(x − 3)] becomes 1/2/(x − 1) − 4/(x − 2) + 9/2/(x − 3). The coefficient list may contain negative values and proper fractions; neither is an error.
Scope, limits, and troubleshooting
This page supports only proper fractions whose denominator is already supplied as two to four different factors of the form x − r, with integer r. It does not factor a polynomial, divide an improper fraction, or handle a repeated root such as (x − 1)². It also excludes irreducible quadratic factors and symbolic parameters. Use polynomial division or a computer algebra system when those structures are required.
Each entered integer is limited to an absolute value of 1,000,000. A duplicate-root message means the denominator needs repeated-factor terms that this focused calculator does not implement. A count message usually means there is only one root, more than four roots, or the numerator degree is not below the denominator degree. Empty comma items, decimals, fractions, variables, and grouping separators are rejected.
Using the output
Copy result includes the decomposition, ordered coefficients, and recombination check. It is suitable for notes or for checking a hand calculation, but it is plain text rather than MathML or a CAS command. Keep the exact fractions when integrating or comparing coefficients; converting early to decimals can hide an identity error. Always retain the stated denominator factors because the coefficient order follows the root order you entered.
Frequently asked questions
Why must the roots be distinct?
A repeated factor needs several powers of the same denominator and a different coefficient system.
Can I enter the full denominator polynomial?
No. Enter its known integer roots; polynomial factoring is outside this tool.
Why are there leading zeros in the check?
They align the recovered numerator with every power below the denominator degree.
Are the coefficients rounded?
No. They are reduced exact fractions backed by integers.
Does the result prove equality?
The tool reconstructs and compares every numerator coefficient exactly for the supplied factors.