Polynomial Long Division

Polynomial Long Division Calculator

Q: How does polynomial long division work?

Exactly like numeric long division, with degrees playing the role of place value: divide leading terms (x³ ÷ x = x²), multiply the divisor by that term, subtract, bring down, repeat. Example: (x³ − 4x + 6) ÷ (x − 1) = x² + x − 3 remainder 3.

Q: Why do I need to add zero coefficients for missing terms?

The subtraction step aligns terms by degree, just as numeric long division aligns digits by place value. Writing x³ − 4x + 6 without the 0x² is like writing 4,006 as 4-6: the x² column has to exist to receive the subtraction. This is the most common source of wrong answers by hand.

Q: What does a remainder of 0 mean?

The divisor is a factor of the dividend. That's the factor theorem in action: (x − c) divides P(x) with remainder 0 exactly when P(c) = 0, so long division doubles as a root-verification tool when factoring cubics and quartics.

Q: When should I use synthetic division instead?

Synthetic division is a shortcut that only works when the divisor is linear with leading coefficient 1, i.e. (x − c). For anything else — (2x + 3), (x² + 1) — long division is the general algorithm. This calculator handles both cases through long division.

Q: How is the remainder written in the final answer?

As a fraction over the divisor: dividend/divisor = quotient + remainder/divisor. So (x³ − 4x + 6)/(x − 1) = x² + x − 3 + 3/(x − 1). In calculus this rewrite is how improper rational functions are prepared for integration.

Q: What's the connection to regular long division?

Substitute x = 10 and polynomial division becomes base-10 division: x³ − 4x + 6 at x = 10 is 966, x − 1 is 9, and 966 ÷ 9 = 107 r 3 — matching x² + x − 3 (= 107 at x = 10) remainder 3. Our long division calculator shows the numeric version with the same step structure.

Divide one polynomial by another with the complete long-division work shown — quotient, remainder, and every subtract-and-bring-down step.

100% FreeQuotient + RemainderEvery Step Shown

Enter Polynomials as Coefficients

Highest degree first, zeros included for missing terms — x³ − 4x + 6 is 1, 0, -4, 6

Dividend (numerator)

= x^3 − 4x + 6

Divisor (denominator)

= x − 1

A Full Worked Example, Step by Step

Divide x³ + 0x² − 4x + 6 by x − 1. Step 1: x³ ÷ x = x²; subtract x²(x − 1) = x³ − x² to get x² − 4x + 6. Step 2: x² ÷ x = x; subtract x(x − 1) = x² − x to get −3x + 6. Step 3: −3x ÷ x =−3; subtract −3(x − 1) = −3x + 3 to get 3. Degree 0 < degree 1, so we stop: quotient x² + x − 3, remainder 3. Check by multiplying back: (x − 1)(x² + x − 3) + 3 reproduces the original — the same multiply-and-verify habit that catches sign errors on exams.

Long Division vs. Synthetic Division

Aspect	Long division	Synthetic division
Divisor allowed	Any polynomial	Only x − c
What you write	Full terms with variables	Coefficients only
Speed	Slower, transparent	Fast, error-prone signs
Typical use	Rational functions, calculus rewrites	Root testing, factor theorem

Where Polynomial Division Actually Shows Up

Three places dominate: factoring cubics (find one root by trial, divide it out, solve the remaining quadratic with the quadratic formula); calculus (an improper rational function must be divided before integrating — ∫(x³ − 4x + 6)/(x − 1)dx starts with exactly the division above); and oblique asymptotes (the quotient x² + x − 3 is the end-behavior curve of the rational function). For the numeric analogue with the same bring-down structure, see our long division calculator.

Frequently Asked Questions

How does polynomial long division work?

Exactly like numeric long division, with degrees playing the role of place value: divide leading terms (x³ ÷ x = x²), multiply the divisor by that term, subtract, bring down, repeat. Example: (x³ − 4x + 6) ÷ (x − 1) = x² + x − 3 remainder 3.

Why do I need to add zero coefficients for missing terms?

The subtraction step aligns terms by degree, just as numeric long division aligns digits by place value. Writing x³ − 4x + 6 without the 0x² is like writing 4,006 as 4-6: the x² column has to exist to receive the subtraction. This is the most common source of wrong answers by hand.

What does a remainder of 0 mean?

The divisor is a factor of the dividend. That's the factor theorem in action: (x − c) divides P(x) with remainder 0 exactly when P(c) = 0, so long division doubles as a root-verification tool when factoring cubics and quartics.

When should I use synthetic division instead?

Synthetic division is a shortcut that only works when the divisor is linear with leading coefficient 1, i.e. (x − c). For anything else — (2x + 3), (x² + 1) — long division is the general algorithm. This calculator handles both cases through long division.

How is the remainder written in the final answer?

As a fraction over the divisor: dividend/divisor = quotient + remainder/divisor. So (x³ − 4x + 6)/(x − 1) = x² + x − 3 + 3/(x − 1). In calculus this rewrite is how improper rational functions are prepared for integration.

What's the connection to regular long division?

Substitute x = 10 and polynomial division becomes base-10 division: x³ − 4x + 6 at x = 10 is 966, x − 1 is 9, and 966 ÷ 9 = 107 r 3 — matching x² + x − 3 (= 107 at x = 10) remainder 3. Our long division calculator shows the numeric version with the same step structure.

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