The Solution
Calculation: Polynomial Long Division Calculator
Polynomial Long Division Calculator
Enter your polynomial coefficients above and click "Perform Long Division" to see results.Graph will appear here after you solve.
Cubic Equation Solver
Polynomial Long Division solver
Find real and complex roots, discriminant, and cubic graphs in seconds with our precision solver.
The Equation
Provide your cubic's coefficients and a divisor value r to execute full polynomial long division by (x − r).
f(x) = 0 = 0
The Treatise
What isPolynomial Long Division Calculator?
- For a school student: Polynomial long division works exactly like the long division you learned with numbers. Divide the first terms, multiply back, subtract, bring down the next term, and repeat. The only difference is that the "digits" are powers of x instead of powers of 10.
- For a college student: For a dividend P(x) and divisor D(x), the method finds the unique quotient Q(x) and remainder R(x) with P = DQ + R and deg R < deg D. Each cycle kills the current leading term: divide leading terms, multiply the whole divisor, subtract, and continue with what remains.
- For the advanced reader: The division algorithm is what makes F[x] a Euclidean domain. That single fact makes F[x] a Euclidean domain, opening the door to the Euclidean algorithm for polynomial GCDs, which detects repeated factors (a common factor of P and P′ signals a multiple root), and underpins unique factorization and the theory behind Gröbner bases.
- Why it matters in practice: It is the universal divider: unlike synthetic division, it handles any divisor, linear or not, monic or not. Use it to verify a guessed factor in the factorization calculator, or to confirm a root found via the rational root theorem truly divides the polynomial.
The Formula
Formula / Method
- General form: P(x) / D(x) = Q(x) + R(x) / D(x), with deg R < deg D. Equivalently, P(x) = D(x)Q(x) + R(x).
- Shortcut: When the divisor is (x − c), synthetic division performs the identical computation with far less writing. Reserve long division for everything else.
- Reading the result: Stop when the remaining polynomial has lower degree than the divisor; that leftover is the remainder. A zero remainder proves the divisor is a factor, by the factor theorem generalized.
- A common trap: Both polynomials must be written in descending order with zero placeholders for missing powers. Dividing x³ + 2 without writing x³ + 0x² + 0x + 2 misaligns every column.
The Method
How To Use
- Step 1. Enter the dividend and divisor coefficients in descending order, inserting 0 for any missing power.
- Step 2. Press "Divide" and follow each cycle: divide leading terms, multiply back, subtract, bring down.
- Step 3. Read the quotient from the top row and the remainder from the final leftover.
- Step 4. Verify with (Divisor × Quotient) + Remainder, which must reproduce the dividend exactly.
Key Features
- Shows every subtract-and-bring-down cycle in full, nothing skipped.
- Inserts zero placeholders automatically so columns stay aligned.
- Highlights the remainder and checks that its degree is below the divisor's.
- Reconstructs (Divisor × Quotient) + Remainder as a built-in correctness proof.
- Suggests synthetic division automatically when the divisor is linear.
The Illustration
Visual Diagram
Example Concept
Divide x³ − 6x² + 11x − 6 by (x − 1).
x³ ÷ x = x². Multiply: x²(x − 1) = x³ − x². Subtract: −5x² + 11x. Bring down.
−5x² ÷ x = −5x. Multiply: −5x(x − 1) = −5x² + 5x. Subtract: 6x − 6.
6x ÷ x = 6. Multiply: 6(x − 1) = 6x − 6. Subtract: 0.
Quotient x² − 5x + 6, remainder 0. Check: (x − 1)(x² − 5x + 6) = x³ − 6x² + 11x − 6, the original dividend.
The Deep Dive
Interactive Deep Dive
The algorithm always terminates, and the reason is delightfully simple: each cycle eliminates the current leading term, so the degree of the running remainder strictly decreases. A strictly decreasing sequence of whole numbers cannot continue forever, so the process must stop. It stops exactly when the remainder's degree falls below the divisor's degree, which is the defining property deg R < deg D. Termination is therefore not luck but arithmetic. The quotient and remainder are also unique: no two different pairs can satisfy the same division, which is why checking an answer by reconstruction is decisive.
The degree condition on the remainder is doing real work. When the divisor is (x − c), the remainder must have degree less than 1, meaning it is a constant. That constant is P(c), which is the remainder theorem emerging as a special case of long division. For a quadratic divisor, the remainder can be linear, which is why partial fraction decompositions have the shapes they do.
Synthetic division and long division perform identical arithmetic for linear divisors; synthetic merely omits the powers of x and folds each subtraction into an addition of the negated product. The rule of thumb: linear monic divisor goes to synthetic, everything else goes to long division. Knowing both means never doing more writing than the problem demands.
Dividing out each discovered linear factor in turn is the deflation chain that powers the factorization calculator. For a cubic, two divisions reduce the problem to a quadratic, and the quadratic formula finishes it. Long division is the general tool in this chain because intermediate divisors are not always linear: dividing by a known quadratic factor works the same way.
Repeated long division computes polynomial GCDs via Euclid's algorithm: gcd(P, Q) = gcd(Q, R) where R is the remainder of P ÷ Q. A nontrivial gcd(P, P′) reveals repeated factors, since a multiple root of P is also a root of its derivative. For example, P = x³ − 3x + 2 = (x − 1)²(x + 2) has P′ = 3x² − 3, and Euclid's algorithm on the pair exposes their shared factor (x − 1). This is the computational route to detecting the Δ = 0 case that the discriminant calculator classifies.
The Applications
Real-World Applications
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Partial Fractions
Before decomposing a rational function, divide when the numerator degree meets or exceeds the denominator degree. The polynomial part comes out front and the remainder becomes a proper fraction ready for decomposition. Skipping this step forces the partial fraction machinery to handle an improper form it was never designed for.
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Transfer Functions
Control engineers routinely reduce high-order transfer functions by dividing out known pole factors. Each division lowers the order by the divisor degree while the remainder tracks what was removed, keeping the reduced model faithful to the original.
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Factor Verification
Suspect that (x - 2) is a factor? Divide by it. A zero remainder is proof, a nonzero remainder is a verdict, and either way the question is settled without guessing. Pair this with candidate lists from the rational root theorem for a complete hand-solving workflow.
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Asymptote Analysis
Dividing numerator by denominator of a rational function exposes its end behavior: the quotient is the oblique or horizontal asymptote that the graphing tool approaches at the extremes.
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Simplifying Rational Models
Physics and engineering models often arrive as ratios of polynomials. Division separates the polynomial behavior from the genuinely fractional part, making approximation and analysis far cleaner.
The Cautions
Common Mistakes to Avoid
MISTAKE Nº 1
Misaligning terms by degree: Every term must sit in its degree column. One shifted term corrupts all subsequent subtractions, so align first and compute second.
MISTAKE Nº 2
Subtraction sign errors: The step is subtract, not add. The classic blunder is forgetting to flip every sign of the multiplied-back row before combining.
MISTAKE Nº 3
Stopping too early or too late: Stop exactly when the running remainder has lower degree than the divisor. Continuing further invents fractional terms that do not belong.
MISTAKE Nº 4
Skipping zero placeholders: Missing powers still own columns. Write 0x² explicitly, or the columns collapse and each cycle divides the wrong terms.
MISTAKE Nº 5
Swapping dividend and divisor: P ÷ D is not D ÷ P. If the divisor has higher degree than the dividend, the quotient is simply 0 and the remainder is the dividend itself.
The Reference
Quick Reference Table
Quick Reference TableREF
FormulaP(x) = Q(x) · D(x) + R(x)
Divisor DegreeAny degree (not limited to linear)
Stops Whendeg(R) < deg(D)
AdvantageHandles quadratic and higher divisors
VerificationQ(x)·D(x) + R(x) must equal P(x)
The Questions
Frequently Asked Questions
When do I need long division instead of synthetic? +
Whenever the divisor is not of the form (x - c): quadratic divisors, higher-degree divisors, or non-monic linears that you prefer not to adjust.
What does a zero remainder mean? +
The divisor is a genuine factor. Feed it to the factorization calculator to continue breaking the polynomial down.
What does a nonzero remainder mean? +
The divisor is not a factor. When the divisor is (x - c), that remainder still equals P(c) by the remainder theorem, so the work was not wasted: you evaluated the polynomial at c.
Must the terms be in descending order? +
Yes, with zero placeholders for missing powers. Order and alignment are what make the column arithmetic valid.
What if the divisor has higher degree than the dividend? +
Then the quotient is 0 and the remainder is the dividend itself. No computation needed.
How do I check my answer? +
Compute (Divisor × Quotient) + Remainder and confirm it reproduces the dividend term for term. Because the quotient-remainder pair is unique, a successful reconstruction is a complete proof of correctness.
Is this the same arithmetic as synthetic division? +
For linear divisors, yes: synthetic division is the same computation with the powers of x stripped out. The quotient and remainder are identical; only the bookkeeping differs.
Can coefficients be fractions? +
Yes. Clearing denominators first keeps the arithmetic tidier, but the algorithm works over any field.