Polynomial Long Division Calculator
Divide any two polynomials and get the quotient and remainder with every long division step shown, exact fractions, and a check that P = D·Q + R.
Polynomial Long Division Calculator
Divide polynomials using long division. Enter the dividend (the polynomial being divided) and the divisor (the polynomial you're dividing by). The calculator will show the quotient, remainder, the long-division layout and step-by-step working, using exact fractions (switch to decimals under Display Options).
Polynomial Division
Use ^ for powers, * for multiplication (e.g., x^3 + 2x^2 - 5x + 6)
Use ^ for powers, * for multiplication (e.g., x - 1)
Input Method
Polynomial long division uses a stricter alignment rule than numerical division. You must keep every term's exponent column aligned throughout the entire process. A single misaligned term, especially a missing term you skipped, breaks every subsequent subtraction. Use this polynomial long division calculator to get the exact quotient and remainder with step-by-step working you can copy or check against your own work. It handles missing terms, fractional coefficients, and non-monic divisors using exact rational arithmetic, never rounding.
- Core Formula: Dividend = Divisor × Quotient + Remainder
- Remainder Degree Rule: Remainder degree must be strictly less than divisor degree
- Input Methods: Full expression (x^3+2x-1) or coefficient list (1,0,2,-1)
- Output Precision: Exact fractions by default; switchable to decimals up to 4 places
- Verification: Calculator multiplies quotient by divisor, adds remainder, and confirms original dividend
How to Enter Polynomials: Expression vs Coefficients
The calculator accepts two input formats. Use expression mode when you have the full polynomial text: type x^3 + 2x^2 - 5x + 6 using ^ for powers and * for multiplication (optional between coefficient and variable). Use coefficient mode when you already have the numbers: enter coefficients from highest degree down to constant term, separated by commas. For example, the dividend x^3 + 2x^2 - 5x + 6 becomes 1, 2, -5, 6. The divisor x - 1 becomes 1, -1.
Missing Terms Must Be Represented
A missing term is a term with coefficient zero, and you must write it explicitly as a placeholder. Dividing x^3 + 1 by x - 1 requires you to enter x^3 + 0x^2 + 0x + 1 in expression mode, or 1, 0, 0, 1 in coefficient mode. If you skip the zero placeholders, the calculator misaligns terms and produces a wrong quotient. This is the single most common error students make, the algorithm has no way to guess that a term is missing.
Choosing the Variable
Under Display Options you can change the variable name from x to y, t, or z. The calculator uses that variable throughout the output. If you enter a different symbol in the expression (like using s instead of the selected variable), the calculator returns an error message identifying the mismatch.
Reading the Result: Quotient, Remainder, and Q + R/D Form
The calculator displays three outputs. The quotient (Q(x)) is the polynomial result above the division bracket. Its degree equals the dividend's degree minus the divisor's degree. The remainder (R(x)) is what is left after dividing as much as possible, and its degree is always strictly less than the divisor's degree. If the remainder is zero, the divisor is a factor of the dividend, this is the Factor Theorem in action.
The result is also shown in Q + R/D form: quotient plus remainder over divisor. This is the form you use for rational function analysis, partial fractions in calculus, and for finding oblique asymptotes. Never write the remainder as a decimal, that destroys the exact algebraic relationship. The calculator outputs remainders as exact fractions or polynomials, never as a single decimal number.
Below the main result, a verification section shows the Division Algorithm identity: Dividend = (Divisor × Quotient) + Remainder. The calculator multiplies, adds, and confirms equality exactly. If the verification fails (which happens only if the input contains a syntax error), the calculator halts with an error message before showing any result.
Worked Example With Every Step Laid Out
Divide: x³ + 2x² − 5x + 6 ÷ x − 1
Step 1: Set up. Write the dividend and divisor in descending order. Both are already in order.
Step 2: Divide leading terms. x³ ÷ x = x². Write x² above the bracket as the first term of the quotient.
Step 3: Multiply. Multiply the divisor (x − 1) by x²: x³ − x².
Step 4: Subtract. (x³ + 2x² − 5x + 6) − (x³ − x²) = 3x² − 5x + 6.
Step 5: Bring down. The next term is already in place. Divide the new leading term 3x² by x: 3x. Add to the quotient: x² + 3x.
Step 6: Multiply and subtract. Multiply divisor by 3x: 3x² − 3x. Subtract: (3x² − 5x + 6) − (3x² − 3x) = −2x + 6.
Step 7: Repeat. Divide −2x by x: −2. Add to quotient: x² + 3x − 2. Multiply divisor by −2: −2x + 2. Subtract: (−2x + 6) − (−2x + 2) = 4.
Step 8: Stop. The remainder 4 has degree 0, which is less than the divisor's degree 1. Result: Quotient = x² + 3x − 2, Remainder = 4. In Q + R/D form: x² + 3x − 2 + 4/(x − 1).
The calculator shows each subtraction as a separate line in a long-division layout table, with the divisor written to the left of the bracket and each intermediate remainder aligned below. Use the Show step-by-step division checkbox to toggle this display.
| Feature | Long Division | Synthetic Division |
|---|---|---|
| Divisor type allowed | Any polynomial (linear, quadratic, cubic, etc.) | Only linear binomials of the form (x − c) |
| Setup complexity | Full polynomial terms, aligned by exponent | Uses only coefficients, no variables written |
| Visual layout | Bracket-and-bar layout, each subtraction shown | Compact table with rows for coefficients |
| Handling missing terms | Insert explicit zero-coefficient placeholders | Insert zero in the coefficient row for each missing power |
| Remainder output | Polynomial (may be constant or higher-degree term) | Number only; remainder is P(c) by the Remainder Theorem |
| Works for non‑monic divisor (e.g., 2x−3) | Yes, directly | No, unless you factor out the leading coefficient first |
When to Use Long Division vs Synthetic Division
Use polynomial long division when the divisor has degree 2 or higher, such as x² + 1 or 2x³ − x + 5. Synthetic division only works for linear divisors of the form (x − c). If your divisor is 2x − 3, you cannot apply synthetic division directly. You can factor out the 2 first (giving x − 1.5), perform synthetic division on that, then divide the resulting quotient by 2, but that adds error-prone steps. Long division handles 2x − 3 in one pass without transformation.
The Remainder Theorem states that dividing P(x) by (x − c) gives remainder P(c). Use this when you only need the remainder and the divisor is linear, it is faster than performing the full division. The Factor Theorem follows: if P(c) = 0, then (x − c) is a factor. Neither theorem replaces the division itself; they are shortcuts for specific situations.
For rational functions, polynomial long division reveals slant asymptotes (oblique asymptotes) when the numerator's degree is exactly one more than the denominator's degree. The quotient line (ignoring the remainder) is the asymptote equation. OpenStax 'Precalculus 2e' (Section 3.5) covers this method in detail.
When to Skip Both Methods
If you only need to multiply or add polynomials, use direct expansion or combining like terms, not division. If you are using a computer algebra system and do not need the intermediate steps, a CAS gives the quotient and remainder instantly. The value of long division is seeing how the terms interact, which is essential for understanding why the quotient and remainder take their forms.
Common Failure Mode: What Most Often Goes Wrong
The single most frequent error in polynomial long division, and the one this calculator is designed to catch, is misalignment caused by skipped missing terms. A student who forgets to write 0x² when dividing x³ + 1 by x − 1 will subtract the product from the wrong column and get a quotient that looks plausible but is wrong. The calculator always inserts zero placeholders automatically, but only if you provide them in your input. If you omit them in coefficient mode, the calculator treats the list as complete from highest degree down, which shifts every term. Check your input by looking at the displayed dividend and divisor before you read the results.
A second common error: treating the remainder as a single decimal number. On this calculator, remainders are always exact polynomials or fractions. If you are writing the answer into homework, use the Q + R/D form, never a decimal approximation. A decimal remainder in an algebra assignment is marked wrong because it loses the structure needed for the next step, whether that step is factoring, finding asymptotes, or integrating.
The calculator's step-by-step display shows each subtraction as a separate line in a long-division layout table. Use this display to trace your own work: line up each intermediate remainder with the column of the exponent it belongs to. If a term appears in the wrong column, you have a misalignment error in your input or your manual calculation.
Common Questions
Can this calculator handle polynomials with fractional coefficients?
Yes. The calculator uses exact rational arithmetic (BigInt fractions) throughout. Enter 1/3 as 1/3 or 0.3333; the calculator keeps it as 1/3 and performs all operations exactly. Decimals are converted to fractions internally, so you never lose precision.
What does a zero remainder tell me?
A zero remainder means the divisor divides the dividend exactly. By the Factor Theorem, if the divisor is (x − c), then c is a root of the dividend and (x − c) is a factor. For a non-linear divisor, exact division means the divisor is a polynomial factor of the dividend, which you can use for further factoring.
Why does the calculator show the remainder as a fraction over the divisor?
The Q + R/D form is the standard way to write a division result in algebra. A decimal remainder would lose the exact relationship between the polynomial terms. The fraction form is what you need for partial fraction decomposition, asymptote analysis, and any situation where the remainder must retain its polynomial identity.
Can I use coefficient mode for polynomials with missing terms?
Yes, but you must include zeros for every missing power. For example, x^4 + 1 (with x^3, x^2, and x terms missing) requires 1, 0, 0, 0, 1 as the coefficient list. The calculator interprets the list from highest degree to constant term, and every comma represents one power. Missing a zero shifts all lower coefficients by one column and gives a wrong quotient.
How do I verify the calculator's result by hand?
Multiply the quotient by the divisor, then add the remainder. The result must equal the original dividend exactly. The calculator performs this verification automatically and displays it. If you get a different result by hand, check for sign errors in subtraction and ensure you aligned terms by exponent, not by visual column position.
