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Polynomial Long Division Calculator

Divide a polynomial from two coefficient lists, then see the quotient, remainder, and a direct check that recovers the dividend.

Comma-separated, e.g. 1, 0, 2, 5 for x³ + 2x + 5.

Comma-separated, e.g. 1, 1 for x + 1.

Result

x^2 - x + 3 R 2

x^3 + 2x + 5 divided by x + 1 is x^2 - x + 3 remainder 2

Quick Answer

A polynomial long division calculator splits one polynomial by another. The tool gives a quotient and any remainder. Enter coefficients from the highest power to the constant, with 0 for each missing power. For [1, 0, 2, 5] ÷ [1, 1], the quotient is x² − x + 3 and the remainder is 2.

What Polynomial Long Division Does And How It Works

Polynomial long division repeats a short set of steps. Divide the lead terms. Put the answer in the quotient. Multiply the divisor by that new term. Subtract, bring down the next term, and repeat. Stop when the remainder has less degree than the divisor. The tool takes number lists, not x terms. Each list runs from the top power to the constant. Use 0 when a power is absent. Thus x³ + 2x + 5 uses 1, 0, 2, 5. The zero holds the x² place. This row keeps each term in place. The row also cuts the risk of a text input error. Check the rows before you run the tool. A one-place shift changes each term after the gap.

Polynomial Long Division: Formula Behind The Result

The polynomial long division algorithm verifies the quotient and remainder using the fundamental division relation.

  • Dividend = Divisor × Quotient + Remainder.
  • P(x) = D(x)Q(x) + R(x).

P(x): the dividend polynomial.

D(x): the nonzero divisor polynomial.

Q(x): the quotient polynomial.

R(x): the remainder polynomial.

The degree of R(x) must be lower than the degree of D(x), unless the remainder is 0.

Using Coefficient Lists For Polynomial Long Division

Inputs

  • Dividend coefficients: enter numbers from highest degree to constant term, separated by commas.
  • Divisor coefficients: use the same descending-degree order, separated by commas.
  • Missing powers: enter 0 as a placeholder in the coefficient list.
  • Separator: place a comma between each coefficient.

Steps

  1. Write both polynomials in descending powers of x.
  2. Insert a zero for each missing power.
  3. Enter the dividend and divisor as comma-separated coefficient lists.
  4. Calculate to see the quotient and remainder.
  5. Check the result with Divisor × Quotient + Remainder.

Polynomial Long Division In Practice: A Full Example

Dividend x³ + 2x + 5, divisor x + 1. Coefficient lists: dividend [1, 0, 2, 5], divisor [1, 1].

  1. Divide leading terms: x³ ÷ x = x².
  2. Multiply: x²(x + 1) = x³ + x².
  3. Subtract: (x³ + 0x² + 2x) − (x³ + x²) = −x² + 2x.
  4. Divide leading terms: −x² ÷ x = −x.
  5. Multiply: −x(x + 1) = −x² − x.
  6. Subtract: (−x² + 2x) − (−x² − x) = 3x.
  7. Bring down 5 to get 3x + 5.
  8. Divide leading terms: 3x ÷ x = 3.
  9. Multiply: 3(x + 1) = 3x + 3.
  10. Subtract: (3x + 5) − (3x + 3) = 2.

The quotient is x² − x + 3 and the remainder is 2. (Checked: (x + 1)(x² − x + 3) + 2 = x³ + 2x + 5). No rounding is used because all coefficients are exact integers.

Where Polynomial Long Division Is Most Useful In Algebra

Use long division to reduce rational forms, test factors, and check algebra work. Synthetic division is shorter for x − c. Long division works with more divisor forms. The same steps work by hand.

For other algebraic factoring and polynomial calculators, see the math calculators hub and look up concepts in the glossary.

Assumptions

  • Coefficients follow descending degree order.
  • A zero marks each missing power.
  • The divisor is not zero.
  • Dividend degree is at least divisor degree for a nonzero quotient.
  • Exact values are kept when the input allows.

Limitations

  • The input fields accept coefficients, not expressions such as x^2 + 3x.
  • A missing zero placeholder changes the polynomial and the answer.
  • Decimal coefficients may lead to rounded display values.
  • The calculator does not factor the quotient or find roots.

In Practice

A missing zero is the main input error. Write each power from the top degree to x⁰. Then copy the number in each slot. Keep a 0 in every blank slot.

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Related Guides

Polynomial Long Division Questions With Clear Answers

Why Must Missing Polynomial Terms Use A Zero?

A zero holds the place of any missing power of x in the list. Without that zero, each later number moves to the wrong power. The list then means a new polynomial. This keeps all later terms aligned.

When Does Polynomial Division Have No Remainder?

Division has no remainder when the divisor is a factor of the dividend. The last subtraction gives 0. The quotient then multiplies back to the dividend. The check has no leftover term at all.

How Can I Check A Polynomial Division Result?

Check polynomial division by multiplying the divisor by the quotient, then adding the remainder. A correct check gives the old dividend. Each power of x must have the same number. The check should match term by term.

Is Synthetic Division The Same As Long Division?

Synthetic division is a short coefficient method for a divisor of the form x − c. Long division works with linear or higher-degree divisors as well. Both methods should give the same quotient for x − c.

What If The Divisor Degree Is Larger?

The quotient is 0 when divisor degree is larger than dividend degree. The full dividend stays as the remainder. No division step can remove its lead term in that case.

Sources

Last updated: . Reviewed for accuracy against the formula shown above.