Math
Polynomial Division Calculator
Type the dividend and divisor and the polynomial division calculator divides them the way you would on paper.
Synthetic division shows the full three-row tableau for a linear divisor (fractions and non-monic divisors such as 2x - 1 included); long division shows each divide-multiply-subtract round for any divisor. You get the quotient, remainder, the result written as quotient + remainder/divisor, a check by multiplying back, and the remainder theorem value.
Missing powers are filled in with 0 automatically. Use ^ or superscripts for exponents.
For synthetic division the divisor must be linear, such as x - 2, x + 3, or 2x - 1.
Try an example
Result
Quotient
—
- Remainder
- —
- Quotient + remainder/divisor
- —
- Divides evenly?
- —
- Check
- —
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Study path
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What polynomial division gives you
Dividing polynomials works like dividing whole numbers: dividend = divisor × quotient + remainder, where the remainder has a smaller degree than the divisor. A remainder of 0 means the divisor is a factor, which is how you break a cubic or quartic into smaller pieces once you know one root.
Synthetic division vs long division
| Method | Works for | What you write |
|---|---|---|
| Synthetic division | Linear divisors x - r (also ax - b after dividing the quotient row by a) | Three rows of numbers: coefficients, products, sums |
| Long division | Any divisor, including x^2 + 1 or 2x^2 - x + 3 | Repeated divide, multiply, subtract, bring down |
How to read the synthetic tableau
- Write r (the zero of the divisor) on the left and the dividend's The number multiplying a variable, like the 3 in 3x. on the top row, inserting 0 for any missing power.
- Bring down the first coefficient. Multiply it by r and write the product under the next coefficient.
- Add down the column, then repeat multiply-and-add across the row.
- The bottom row gives the quotient's coefficients, one degree lower than the dividend; the last number is the remainder.
Common mistakes
- Forgetting the 0 placeholder for a missing power, which shifts every column.
- Using the wrong sign for r.
- Subtracting incorrectly in long division: distribute the minus sign to every term of the product.
- Stopping too early: keep dividing until the degree of what is left is smaller than the divisor's degree.
Worked examples
Synthetic division with a remainder
(2x^3 - 3x^2 + 4x - 5) ÷ (x - 2)
Quotient
2x^2 + x + 6
Divides evenly (factor found)
(x^3 - 6x^2 + 11x - 6) ÷ (x - 1) leaves remainder 0
Quotient
x^2 - 5x + 6
Missing power and negative zero
(x^4 - 16) ÷ (x + 2)
Quotient
x^3 - 2x^2 + 4x - 8
Non-monic linear divisor
(2x^3 - 3x^2 + 4x - 5) ÷ (2x - 1)
Quotient
x^2 - x + 3/2
Long division by a quadratic
(2x^3 + 3x^2 - x + 5) ÷ (x^2 + 1)
Quotient
2x + 3
Dividend smaller than divisor
(x + 1) ÷ (x^2 + 1) has quotient 0
Quotient
0
Synthetic with a quadratic divisor is rejected
Switch to long division for x^2 + 1
Quotient
—
Division by zero
A divisor of 0 is not allowed
Quotient
—
Frequently asked questions
When can I use synthetic division?+
When the divisor is linear: x - r, x + r, or ax + b. For ax + b, divide by x + b/a and then divide the quotient coefficients by a; the remainder is unchanged. This calculator does that automatically.
What does a remainder of 0 mean?+
The divisor is a factor of the dividend, and the zero of the divisor is a root of the dividend. That is the factor theorem.
What is the remainder theorem?+
Dividing f(x) by (x - r) always leaves the remainder f(r). So synthetic division is also a fast way to evaluate a polynomial at x = r.
Why do I need placeholders for missing powers?+
Each column of the synthetic tableau (and each subtraction in long division) lines up by degree. Writing x^3 - 1 as x^3 + 0x^2 + 0x - 1 keeps the columns aligned.
How do I write the final answer?+
Either as dividend = divisor × quotient + remainder, or as quotient + remainder/divisor. The calculator shows both.
Can the calculator divide by x^2 + 1 or another quadratic?+
Yes. Choose Long division. Synthetic division needs a linear divisor, but long division works for any degree.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026