Polynomial Long Division Calculator

Divide any polynomial by any other polynomial — not just linear divisors — with every subtraction step shown.

(x³−2x²−4) ÷ (x−3)Linear Divisor

(2x³+3x²−8x+3) ÷ (2x−1)Non-Monic Divisor

(x⁴+x³−7x²−x+6) ÷ (x²−1)Quadratic Divisor

(x³+1) ÷ (x+1)Zero Remainder

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Result

Answer

Steps go here…

How This Differs From Synthetic Division

Synthetic division is a shortcut that only works when you’re dividing by something as simple as x − c. The moment your divisor has an x² term, or even just a leading coefficient other than 1, that shortcut stops applying — and you’re back to full polynomial long division.

Long division works exactly like the number division you learned before algebra: divide the leading terms, multiply that result by the whole divisor, subtract, and repeat with what’s left over. It’s slower to do by hand, but it never runs out of cases it can handle.

The repeating cycle

divide \to multiply \to subtract \to bring down

Stop once the remainder’s degree is lower than the divisor’s degree — whatever’s left is the final remainder.

Two Worked Examples

Example 1 — dividing by a quadratic

Divide: x⁴ + x³ − 7x² − x + 6 by x² − 1

Step 1. Divide leading terms: x⁴ ÷ x² = x². Multiply the whole divisor by x²: x⁴ − x². Subtract from the dividend: (x⁴+x³−7x²−x+6) − (x⁴−x²) = x³ − 6x² − x + 6.

Step 2. Divide again: x³ ÷ x² = x. Multiply: x³ − x. Subtract: (x³−6x²−x+6) − (x³−x) = −6x² + 6.

Step 3. Divide again: −6x² ÷ x² = −6. Multiply: −6x² + 6. Subtract: (−6x²+6) − (−6x²+6) = 0.

The remainder’s degree just dropped below the divisor’s degree (in this case, it hit 0 exactly), so we’re done: quotient = x² + x − 6, remainder = 0. This means (x² − 1) divides evenly into the original quartic.

Example 2 — a divisor with a leading coefficient

Divide: 2x³ + 3x² − 8x + 3 by 2x − 1

Step 1. 2x³ ÷ 2x = x². Multiply: x²(2x − 1) = 2x³ − x². Subtract: 4x² − 8x + 3.

Step 2. 4x² ÷ 2x = 2x. Multiply: 2x(2x − 1) = 4x² − 2x. Subtract: −6x + 3.

Step 3. −6x ÷ 2x = −3. Multiply: −3(2x − 1) = −6x + 3. Subtract: 0.

Quotient: x² + 2x − 3, remainder 0. Notice each “divide leading terms” step has to account for the divisor’s own leading coefficient (2, not 1) — that’s the step people usually get wrong when the divisor isn’t monic.

Common Mistakes

Forgetting to subtract, not add. After multiplying the divisor by the new quotient term, you subtract that result from the current remainder — a sign slip here compounds through every later step.
Missing placeholder terms. Just like in synthetic division, a missing power of x in the dividend still needs a coefficient of 0 so nothing gets misaligned.
Stopping too early or too late. Keep going only while the remainder’s degree is still greater than or equal to the divisor’s degree. Stop the moment it drops below.
Ignoring the divisor’s leading coefficient. If the divisor is 2x − 1, each step divides by 2x, not just x — forgetting this throws off every term in the quotient.

Frequently Asked Questions

Use synthetic division only when the divisor is linear with a leading coefficient of 1, in the exact form x – c. For anything else, including a divisor like 2x – 1 or x^2 – 1, polynomial long division is the method that actually applies.

It means the divisor doesn’t evenly divide the dividend. You can still write the result as dividend = (divisor)(quotient) + remainder, the same relationship a plain number division like 7 divided by 2 gives you: 7 = 2(3) + 1.

No. The quotient’s degree is always exactly the dividend’s degree minus the divisor’s degree. If the dividend’s degree is lower than the divisor’s, the division doesn’t proceed at all, the whole dividend is simply the remainder.

Yes. The algorithm doesn’t require whole-number coefficients anywhere, dividing leading terms can produce a fraction or decimal, and the process continues exactly the same way.