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Polynomial Division Calculator

Divide one polynomial by another with long division or synthetic division, and follow every step.

Write powers with ^, like 3x^2, or type the coefficients, like 2, −3, 4, −5. Synthetic division needs a divisor like x − 2 or x + 3.

Quotient and remainder

2x² + x + 6, remainder 7

(2x³ − 3x² + 4x − 5) ÷ (x − 2) = 2x² + x + 6, remainder 7.

Show me how

All steps as text

    Tip: press ← and → to move between steps, or click any step.

    Polynomial long division

    It works like long division with numbers. Repeat until what is left has a lower power than the divisor:

    1. Divide the leading term of what is left by the leading term of the divisor.
    2. Multiply the whole divisor by that term.
    3. Subtract, and bring down the next term.

    For (2x³ − 3x² + 4x − 5) ÷ (x − 2): 2x³ ÷ x = 2x², then x² ÷ x = x, then 6x ÷ x = 6, leaving a remainder of 7. So the answer is 2x² + x + 6 remainder 7.

    Synthetic division

    A shortcut when dividing by x − r. Write r and the coefficients (with 0 for any missing power). Bring down the first coefficient; then, column by column, multiply by r, write the result under the next coefficient and add. The last number is the remainder, and the others are the quotient’s coefficients.

    For x + 3, r = −3. For a divisor like 2x − 1, use long division instead.

    The remainder and factor theorems

    Dividing P(x) by x − r always leaves the remainder P(r). In the example, P(2) = 2·8 − 3·4 + 4·2 − 5 = 7. If the remainder is 0, then x − r is a factor of P(x): for example, x² − 4 ÷ (x + 2) = x − 2 exactly.

    Questions

    What if a power is missing?

    Treat it as 0. For x³ + 1, the coefficients are 1, 0, 0, 1. Forgetting the zeros is the most common mistake in synthetic division.

    When do I stop dividing?

    When what is left has a lower power than the divisor. That leftover is the remainder.

    How do I write the answer?

    As dividend = divisor × quotient + remainder, or as quotient + remainder ÷ divisor. For the example: 2x² + x + 6 + 7/(x − 2).