Last updated: October 2026
How do you divide polynomials? The 4 repeating steps
Short answer: repeat four moves — divide the leading terms, multiply the result by the whole divisor, subtract (flipping every sign), bring down the remaining terms — until the remainder's degree is lower than the divisor's degree.
Polynomial long division is exactly the long division you learned for numbers, with x-powers playing the role of place values. Take (x³ + 2x² − 5x + 1) ÷ (x + 2). The leading term x³ divided by the divisor's leading term x gives x² — that is the first piece of the quotient. Multiply x² back across the divisor to get x³ + 2x², subtract it from the dividend (every sign flips), and you are left with −5x + 1. Now repeat: −5x ÷ x = −5, multiply to get −5x − 10, subtract to get 11. The constant 11 has a lower degree than x + 2, so you stop. Quotient x² − 5, remainder 11.
Worked example: (x³ + 2x² − 5x + 1) ÷ (x + 2)
Short answer: quotient x² − 5, remainder 11 — so the division equals x² − 5 + 11/(x + 2). Every move is shown below.
Step 1 — Divide: x³ ÷ x = x². Write x² in the quotient.
Step 1 — Multiply: x² × (x + 2) = x³ + 2x².
Step 1 — Subtract: (x³ + 2x² − 5x + 1) − (x³ + 2x²) = −5x + 1. Notice the −5x and +1 were simply brought down.
Step 2 — Divide: −5x ÷ x = −5. Append it: quotient so far x² − 5.
Step 2 — Multiply: −5 × (x + 2) = −5x − 10.
Step 2 — Subtract: (−5x + 1) − (−5x − 10) = 11. Watch the double negative: 1 − (−10) = 11.
Done: 11 is a constant and the divisor has degree 1, so 11 is the remainder. Answer: x² − 5 remainder 11.
Run it in the calculator above and compare — the tool prints exactly these rows, so you can check homework line by line instead of only seeing the final answer.
Shortcut: synthetic division when the divisor is x − c
Short answer: if the divisor looks like x − c (for example x − 7, or x + 2 which is x − (−2)), skip the tableau: write the dividend's coefficients in a row, bring down, multiply by c, add — repeat. The last number is the remainder.
Divide (x³ − 12x² + 38x − 17) by (x − 7). Here c = 7. Coefficients: 1, −12, 38, −17. Bring down 1; multiply 1 × 7 = 7, add to −12 → −5; multiply −5 × 7 = −35, add to 38 → 3; multiply 3 × 7 = 21, add to −17 → 4. Bottom row: quotient x² − 5x + 3, remainder 4. Three lines of arithmetic instead of three long-division steps — same answer. The calculator detects this case automatically and shows the synthetic tableau whenever your divisor qualifies. Synthetic division only works for linear divisors with leading coefficient 1; anything else (x² + 1, 2x + 1) needs full long division.
When the remainder is zero: the factor theorem
Short answer: a zero remainder means the divisor is a factor of the dividend — and for a divisor x − c, it means c is a root of the polynomial.
Try (x² − 1) ÷ (x − 1) in the calculator: the steps give quotient x + 1 and remainder 0, so x² − 1 = (x − 1)(x + 1). That is the factor theorem in action: x − c divides a polynomial evenly exactly when plugging in c gives zero. Teachers love this question because it turns division into root-finding — and it is the fastest way to check a factorization you guessed.
Polynomial long division, start to finish: the full guide
Short answer: line the polynomials up by descending powers, run divide–multiply–subtract–bring down until the remainder is too small to divide, then verify with quotient × divisor + remainder. The details that trip people up are below.
Set up correctly. Write both polynomials in descending order of powers, and insert placeholder zeros for any missing power — dividing x³ + 1 by x + 1 goes wrong fast if you forget the invisible 0x² + 0x. The calculator accepts terms in any order and combines like terms, but on paper the placeholders are what keep columns aligned.
The four moves, precisely. Divide: leading term of the working dividend ÷ leading term of the divisor — only the leading terms, never the whole polynomials. Multiply: the new quotient term times the entire divisor. Subtract: subtract that product from the working dividend, distributing the minus across every term — this is where most errors happen (see the FAQ on sign flips). Bring down: carry the untouched lower terms into the next round. Repeat until the working remainder's degree drops below the divisor's degree.
Know when to stop. You stop the moment the remainder's degree is smaller than the divisor's degree — a constant remainder against a linear divisor, a linear remainder against a quadratic divisor, and so on. There is no requirement that the remainder be zero; most divisions leave one.
Common mistakes to avoid. Forgetting to flip signs when subtracting (the #1 error). Dividing by the whole divisor instead of its leading term. Dropping the remainder from the final answer — the complete answer is quotient + remainder/divisor, not just the quotient. And reaching for synthetic division on a divisor like 2x + 1 or x² + 1, where it does not apply.
Always check. Multiply quotient × divisor, add the remainder, and confirm you get the original dividend. It takes one line and catches every arithmetic slip — which is why the calculator above runs this check on every division and shows you the working.
Frequently asked questions
How do you do polynomial long division step by step?
Repeat four moves until the remainder's degree is lower than the divisor's degree: (1) Divide the leading term of the current dividend by the leading term of the divisor and write the result in the quotient. (2) Multiply that term by the whole divisor. (3) Subtract the product from the current dividend — this flips every sign. (4) Bring down the remaining terms. The calculator above performs all four moves and shows each one.
What is the remainder in polynomial division?
The remainder is whatever is left when the divisor no longer fits — formally, the polynomial of degree lower than the divisor that satisfies dividend = quotient × divisor + remainder. For example, (x³ + 2x² − 5x + 1) ÷ (x + 2) leaves remainder 11, so the answer can be written x² − 5 + 11/(x + 2).
When can I use synthetic division instead of long division?
Whenever the divisor is linear with a leading coefficient of 1 — that is, of the form x − c (like x − 7 or x + 2, where c = −2). Synthetic division compresses the whole long-division tableau into one row of arithmetic. This calculator detects that case automatically and shows the synthetic-division shortcut alongside the long division.
What if the divisor is not linear, like x² + 1?
Use ordinary polynomial long division — the same four steps work for any divisor degree. For example, (2x³ + 3x² − x + 5) ÷ (x² + 1) gives quotient 2x + 3 with remainder −3x + 2. Synthetic division cannot handle this case, but the calculator's long-division engine can.
How do I check my polynomial division answer?
Multiply the quotient by the divisor and add the remainder — you must get back the exact dividend: Q × D + R = dividend. The calculator runs this check automatically after every division and shows the multiplication worked out, so a wrong answer is impossible to miss.
Why does subtracting flip all the signs?
Because subtracting a polynomial means adding its opposite: (a − b) − (c − d) = a − b − c + d. The single most common error in polynomial long division is forgetting to distribute the minus sign across the whole product — for instance writing (x³ + 2x²) − (x³ + 2x²) as 4x² instead of 0. Watch the signs on the Subtract row of each step above.
Why learn polynomial long division?
It is the engine behind factoring polynomials, finding roots via the factor theorem (if x − c divides evenly, c is a root), simplifying rational expressions, and partial-fraction decomposition in calculus. Synthetic division is the fast path, but long division works on every divisor — it is the general skill.
