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Numbers, Units & Calculators

Polynomial Long and Synthetic Division Calculator

Divide polynomials with exact fractions, aligned long division, and expanded synthetic division. Check each quotient and remainder.

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Input

Enter terms with one letter, degree at most 50, and coefficients up to 60 digits.

Enter a nonzero polynomial using the same letter, or a constant.

Ready for a result

Divide polynomials with exact fractions, aligned long division, and expanded synthetic division. Check each quotient and remainder.

Worked example result

Select Load example to use these values in the calculator.

Dividend
2x^3 + 3x^2 - x + 5
Divisor
x + 2
Method
both

Quotient

2x^2 - x + 1
Remainder
3
Dividend degree
3
Divisor degree
1
Result
2x^2 - x + 1 + 3/(x + 2)

Long division

                2x^2  - x + 1
      ┌───────────────────────
x + 2 │  2x^3 + 3x^2  - x + 5
       -(2x^3 + 4x^2)
       ──────────────
                -x^2  - x
              -(-x^2 - 2x)
              ────────────
                        x + 5
                      -(x + 2)
                      ────────
                            3

Long division

  1. Divide leading terms to obtain 2x^2. Multiply by 2x^2. Subtract the product. Bring down the coefficient of x^1: -1.
  2. Divide leading terms to obtain -x. Multiply by -x. Subtract the product. Bring down the coefficient of 1: 5.
  3. Divide leading terms to obtain 1. Multiply by 1. Subtract the product. No terms remain to bring down.

Synthetic division

-2 │ 2   3  -1    5
   │    -4   2   -2
   └───────────────
     2  -1   1 │  3

Remainder Theorem: c = -2; f(-2) = 3 = remainder.

Verified: divisor × quotient + remainder = dividend. (x + 2) × (2x^2 - x + 1) + (3) = 2x^3 + 3x^2 - x + 5.

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Long division repeatedly subtracts a divisor multiple. Expanded synthetic division normalizes the divisor and uses its negated lower coefficients.

The result includes quotient, remainder, method working, and a multiplication identity check.

  • Compare long and synthetic division for an algebra assignment.

Divide a cubic polynomial

Input: 2x^3 + 3x^2 - x + 5

Settings: Divisor: x + 2; Method: Both

Result: 2x^2 - x + 1

Both methods give this quotient and remainder 3. Substitution gives f(-2) = 3.

Choose long or synthetic division

  1. Enter Dividend and a nonzero Divisor with the same variable.
  2. Choose Method.
  3. Select Divide polynomials and inspect the working.

Polynomial syntax and exact coefficient bounds

  • Use one single-letter variable. Grouped sums and variable denominators are unsupported.
  • Constant divisors support long division only.
  • A lower-degree dividend produces quotient zero and keeps the dividend as remainder.

Questions

Can coefficients use decimals or fractions?

Yes. The calculator converts decimal and rational coefficients to exact fractions.