Polynomial Long Division Calculator

Divide any two polynomials with clear, step‑by‑step working. Enter expressions using any variable, decimals or fractions, and missing terms. See the quotient, remainder, synthetic division for linear divisors, and a fully formatted division tableau. Helpful for algebra practice, checking homework, or building intuition about degrees, leading terms, and coefficient arithmetic. Supports sign errors detection and optional rounding for cleaner presentation.

Inputs

Tips

  • Write polynomials like 3x^4 - 2x + 7. Missing terms are handled automatically.
  • Coefficients can be integers or decimals.
  • Synthetic division is available when the divisor is linear of the form x - r.

Results

Problem
Quotient
Remainder
Long‑division Layout
Quotient:
Long‑Division Steps
Synthetic Division (when divisor is x − r)

How Polynomial Long Division Works

  1. Order both polynomials by descending powers of x. Fill any missing powers with coefficient 0.
  2. Divide leading terms: divide the leading term of the dividend by the leading term of the divisor to get the next term of the quotient.
  3. Multiply & subtract: multiply the entire divisor by that term and subtract from the current dividend to get a new remainder.
  4. Repeat until the remainder’s degree is less than the divisor’s degree. That remainder is your final remainder.
  5. If the divisor is linear and monic (x - r), you can use synthetic division to streamline the arithmetic.

Built for clarity and learning: exact arithmetic on coefficients you provide, automatic term alignment, and beautiful math rendering via MathJax.

Polynomial long division is the algebraic twin of the long division you learned with numbers. You repeatedly match leading terms, multiply, subtract, and bring down the next term until the remainder has degree less than the divisor. This guide shows the full recipe, explains why it works, and illustrates it with carefully chosen examples—including a case with missing terms, and another with a non‑zero remainder. By the end, you’ll be able to set up and complete divisions reliably and check your answers.

When should I use it?

  • To simplify a rational expression like P(x)/Q(x) where deg P ≥ deg Q.
  • To separate a function into a polynomial plus a proper fraction (useful before integration or limits).
  • To factor a polynomial when you know or suspect a factor (e.g., try x − a using the Remainder Theorem).
  • As groundwork for partial fractions and for understanding synthetic division.

Vocabulary & Setup

DividendThe polynomial being divided.
DivisorThe polynomial you divide by.
QuotientThe result at the top.
RemainderWhat’s left when you can’t continue (degree < degree of divisor).

Always write polynomials in descending powers, and fill in any missing terms with 0 coefficients. For example, write x4 − 5x2 + 4 as x4 + 0x3 − 5x2 + 0x + 4.

x − 2 (divisor) x³ − 2x² + 4x − 8 (dividend) x² + 4 (quotient)
Set up like numerical long division: divisor on the left of the bracket, dividend inside, quotient on top.

The Core Algorithm (Recipe)

  1. Order & pad: Arrange dividend and divisor in descending powers of x. Insert missing powers with 0 coefficients.
  2. Divide leading terms: Divide the leading term of the current dividend (or remainder) by the leading term of the divisor. That gives the next term of the quotient.
  3. Multiply back: Multiply the entire divisor by that new quotient term; write the result under the aligned powers.
  4. Subtract carefully: Subtract the line you just wrote. Watch the signs—subtracting a polynomial changes every sign.
  5. Bring down next term(s): Continue the process until the degree of what remains is less than the degree of the divisor.
  6. State result: Write Dividend = Divisor × Quotient + Remainder. If the remainder is 0, the divisor is a factor.
Tip: Underline or lightly box the “current” leading term at each cycle to keep your eyes from wandering. This reduces minus‑sign errors and misalignment.

Worked Example 1 (Linear Divisor, Remainder 0)

Divide x3 − 2x2 + 4x − 8 by x − 2.

  1. Leading terms: x3 ÷ x = x2. Write x2 in the quotient.
  2. Multiply back: x2(x − 2) = x3 − 2x2. Subtracting gives 0x3 + 0x2 + 4x − 8.
  3. Next leading terms: 4x ÷ x = 4. Add +4 to the quotient.
  4. Multiply back: 4(x − 2) = 4x − 8. Subtracting yields remainder 0.

Answer: x3 − 2x2 + 4x − 8 = (x − 2)(x2 + 4). So (x − 2) is a factor, and the quotient is x2 + 4.

Why does this work?

Each cycle peels off the highest power by matching the leading terms. Subtracting cancels that power, leaving a smaller-degree remainder. This must finish in finitely many steps because the degree strictly drops.

Worked Example 2 (Quadratic Divisor, Mind the Missing Terms)

Divide x4 + 0x3 − 5x2 + 0x + 4 by x2 − x − 2.

  1. Leading terms: x4 ÷ x2 = x2. Place x2 in the quotient.
  2. Multiply back: x2(x2 − x − 2) = x4 − x3 − 2x2.
  3. Subtract: (x4 + 0x3 − 5x2) − (x4 − x3 − 2x2) = x3 − 3x2; bring down 0x + 4.
  4. New leading terms: x3 ÷ x2 = x. Add +x to the quotient.
  5. Multiply back: x(x2 − x − 2) = x3 − x2 − 2x.
  6. Subtract: (x3 − 3x2 + 0x) − (x3 − x2 − 2x) = −2x2 + 2x; bring down +4.
  7. Leading terms: −2x2 ÷ x2 = −2. Add −2 to the quotient.
  8. Multiply back: −2(x2 − x − 2) = −2x2 + 2x + 4. Subtract to get remainder 0.

Answer: Quotient x2 + x − 2, remainder 0. Indeed, x4 − 5x2 + 4 = (x2 − x − 2)(x2 + x − 2).

Notice the zero placeholders (0x3, 0x). Without them, alignment breaks and subtraction goes wrong.
current: x³ − 3x² + 0x + 4 subtract: (x³ − 1x² − 2x) result: 0x³ − 2x² + 2x + 4
Align like terms vertically so subtraction only touches matching powers.

Worked Example 3 (Remainder Present)

Divide 2x3 + 3x2 − x + 5 by x2 − 1.

  1. Leading terms: 2x3 ÷ x2 = 2x. Write 2x in the quotient.
  2. Multiply back: 2x(x2 − 1) = 2x3 − 2x. Subtract to get 3x2 + x + 5.
  3. Leading terms: 3x2 ÷ x2 = 3. Add +3 to the quotient.
  4. Multiply back: 3(x2 − 1) = 3x2 − 3. Subtract to get remainder x + 8.

Answer: Quotient = 2x + 3, Remainder = x + 8. You can write the final result as 2x + 3 + (x + 8)/(x2 − 1) or simply state the identity Dividend = Divisor × Quotient + Remainder.

Quick Checklist (Prevent Common Errors)

  • ✳️ Polynomials are in descending order and zero‑padded.
  • ✳️ Each quotient term comes from leading ÷ leading.
  • ✳️ Whole divisor is multiplied back, not just its leading term.
  • ✳️ Subtract the entire line (flip every sign).
  • ✳️ Stop once the remainder’s degree is less than the divisor’s degree.

Connections: Synthetic Division & The Remainder Theorem

When the divisor is linear, x − a, you can use synthetic division—a streamlined table method. It produces the same quotient and remainder as long division with far fewer written symbols. The Remainder Theorem says the remainder of dividing P(x) by x − a equals P(a). If P(a) = 0, then x − a is a factor and the remainder vanishes.

DivisorRecommended MethodWhy
Linear (x − a)Synthetic or LongSynthetic is faster but long division shows structure.
Degree ≥ 2Long DivisionWorks universally and helps keep alignment.

Practice Problems

  1. Divide x3 − 4x + 4 by x − 2.
  2. Divide 3x3 + 5x2 − 6x + 1 by x2 + x. (Remember to pad the divisor as x2 + x + 0.)
  3. Divide 2x4 − x2 + 7 by x2 − 3.
  4. Divide x4 − 2x3 + x − 5 by x2 − x − 1.
  5. Divide 4x3 − 8x + 12 by x − 2.
Show Answers
  1. x2 + 2x + 0, remainder 0. (Because x3 − 4x + 4 = (x − 2)(x2 + 2x).)
  2. 3x + 2, remainder −6x + 1.
  3. 2x2 + 6, remainder −x2 + 7.
  4. x2 − x − 1, remainder x − 4.
  5. 4x2 + 8x + 8, remainder 28.

Proof Sketch: Why the Remainder’s Degree Is Smaller

Let P(x) be the dividend and D(x) the divisor with leading term amxm. At each cycle, you choose a quotient term q(x) so that q(x)D(x) has the same leading term as the current polynomial you are reducing. Subtracting cancels that leading term. Because degrees strictly decrease in the sequence, the process stops after finitely many steps with a remainder whose degree is less than m. This is the polynomial analogue of the Euclidean algorithm and underlies the uniqueness of the quotient and remainder.

Frequent Mistakes & How to Fix Them

  • Forgetting zero placeholders: Always pad missing powers. Without them, alignment slips and subtraction corrupts coefficients.
  • Dividing by the wrong term: Only divide leading by leading. Don’t “peek” at inner terms.
  • Partial multiplication back: Multiply the entire divisor by the new quotient term, not just its front.
  • Sign errors in subtraction: Think “add the opposite”. Rewrite the subtraction line with flipped signs before adding.
  • Stopping too early: Continue until the remainder’s degree is strictly less than the divisor’s degree.

Mini‑Checklist for Exams

  • Write both polynomials in descending order and include zeros.
  • Compute each quotient term = leading÷leading.
  • Multiply the divisor and subtract (flip signs).
  • Repeat until the remainder’s degree is smaller.
  • State the final identity P = D·Q + R and, when helpful, express the result as Q + R/D.

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