2026-09-29 9 min

Synthetic Division Calculator

Divide by (x − c) with a synthetic tableau, Remainder Theorem check and factor verdict. Free online calculator.

Calculator

Synthetic Inputs

Dividend coefficients

Highest-degree-first. Results update live.

c — divisor is (x − c) Help

For x + 2 enter −2

Results update live as you type

Status:(x^3 - 6x^2 + 11x - 6) ÷ (x − 1)R = 0 · factor ✓

Synthetic Summary

Dividendx^3 - 6x^2 + 11x - 6

Divisor(x − 1)

Quotientx^2 - 5x + 6

Bottom row1, -5, 6, 0

Results

Optimal

Quotient Q(x)

x^2 - 5x + 6

Bottom row except the last cell

Remainder = P(1)

Remainder Theorem: R = P(c)

Factor Verdict

(x − 1) IS a factor

Factor Theorem: zero remainder ⇔ factor

Direct P(c) Check

Independent power-sum must match

Quotient Degree

Always one below the dividend

Rail arithmetic runs in double precision; the amber remainder is independently confirmed by direct power-sum evaluation.

Visualizations

Bottom-Row Totals

1q0

5q1

6q2

Factor Test

Root ✓100%

|Quotient(c)| vs |Remainder|

|Q|2

Formula

The rail multiplies by c and adds down the line — Horner evaluation and division fused into one sweep.

P(x) = (x − c)·Q(x) + P(c); bottom row gives Q, last cell P(c)

Alternative Form

Per cell: total = previous·c + next coefficient

Variables

Symbol	Name	Unit	Description
P	Dividend	coeffs	Highest-degree-first coefficients
c	Root candidate	number	Divisor x − c; test value
Q	Quotient	coeffs	Bottom row minus last cell
R	Remainder	number	Last cell = P(c) exactly

Explanation

Blue rail box holds c; each row multiplies the running total by c and adds the next coefficient; amber cell is P(c).

Derivation

Long division by monic (x−c) needs only the multiply-add per column — the rail drops the x bookkeeping, identical to Horner.

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Set up: coefficients on top, c outside the rail.

P ÷ (x − c)

P:x^3 - 6x^2 + 11x - 6

divisor:(x − 1)

Result: Divide x^3 - 6x^2 + 11x - 6 by (x − 1)

Step 2

Bring down the leading coefficient untouched.

total₀ = aₙ

first:1

Result: Opens with 1

Step 3

Loop multiply-by-c then add the next coefficient.

total = total·c + aᵢ

c:1

Intermediate: down 1 → 1×1+-6=-5 → -5×1+11=6 → 6×1+-6=0

Result: Bottom row: 1, -5, 6, 0

Step 4

Split the bottom row into quotient and remainder.

Q = all but last; R = last

Q:x^2 - 5x + 6

R:0

Result: Q(x) = x^2 - 5x + 6, R = 0

Step 5

Confirm the Remainder Theorem: R must equal direct P(c).

R = P(c)?

R:0

direct:0

Result: 0 = 0 ✓ theorem holds

Step 6

Deliver the Factor Theorem verdict.

R = 0 ⇔ factor

Result: (x − 1) divides P — deflate and continue hunting

Final Answer

x^2 - 5x + 6

Worked Examples

See how the calculator works with different scenarios.

Scenario

Textbook cubic with root 1.

Inputs

coeffs:1,-6,11,-6

c:1

Calculation

Bottom row 1, −5, 6, 0

Result

Q = x²−5x+6, R = 0 — factor ✓

Interpretation

Deflate to the quadratic and keep factoring.

Understanding Your Result

Your Result: Q(x) = x^2 - 5x + 6, R = P(1) = 0

Typical range: Any real coefficients and c; zero remainder certifies a factor.

What This Means

The quotient is the deflated polynomial; the remainder is the value P(c) — zero means you found a root.

Why It Matters

Synthetic division is the engine of root hunting: test candidates in seconds and deflate to lower degrees.

Professional Interpretation

Chain synthetics: each zero remainder peels one linear factor — repeat until the quadratic formula finishes the job.

Recommendations

•Negate for (x + c) divisors
•Include every zero slot
•Compare R with direct P(c)
•Deflate after each root found
•Fall back to long division otherwise

Educational Content

Introduction

Synthetic division squeezes linear long division into a single rail of multiply-add cells.

Concept Explanation

Because x − c is monic and linear, each long-division cycle reduces to total·c + next — the rail tracks only numbers.

Applications

▸Rational-root testing loops

▸Deflation after each zero

▸Fast P(c) evaluation

▸Factor certification

▸Partial-fraction preparation

▸Classroom division drills

Benefits

•One-row tableau
•Remainder theorem built in
•Factor verdict automatic
•Direct-sum cross-check
•Deflation-ready quotient

Limitations

•Divisors (x − c) only
•Numeric coefficients
•Constants need no tableau

Best Practices

•Write c with correct sign
•Keep all zero slots
•Bring the leader straight down
•Circle the remainder cell
•Verify R against P(c)

Common Misconceptions

Myth: Synthetic works for any divisor.

Truth: Only monic linear x − c; quadratics and general linears need long division.

Myth: For x + 2 use c = 2.

Truth: x + 2 = x − (−2), so c = −2 — the sign flip is mandatory.

Myth: The bottom row is all quotient.

Truth: The last cell is the remainder; quotient is everything before it.

Myth: Non-zero remainders are failures.

Truth: They are values: R = P(c) still evaluates the polynomial correctly.

Practical Tips

•Negate deliberately for (x + c)
•Keep zero slots in the rail
•Bring the leader down first
•Multiply before adding per cell
•Circle the amber remainder
•Compare R with P(c)
•Deflate and repeat on roots

Common Mistakes

Wrong sign on c for (x + c)

Rewrite x + 2 as x − (−2) before starting the rail.

Dropping interior zero coefficients

Zeros are real rail columns; skipping shifts every total.

Counting the remainder as quotient

Quotient stops one cell early — the last cell is R.

Using synthetic for non-linear divisors

Quadratics need long division; synthetic is x − c only.

Ignoring the verification sum

Direct P(c) must equal the amber cell — compare every run.

Assumptions

This calculator makes the following assumptions to provide estimates:

Monic linear divisor: Divisor is exactly (x − c).

Positional coefficients: First number is the highest power.

Real numbers: Coefficients and c are real.

Exact values: Inputs are exact, not measured.

Display rounding: Outputs round to 6 decimals.

Limitations

Be aware of these limitations when using this calculator:

x − c only: General divisors use long division.

Numeric scope: Symbolic parameters unsupported.

Single variable: Univariate polynomials only.

Tolerance edge: Near-zero remainders verdict as roots within 1e-9.

Frequently Asked Questions

Answers to common questions about this calculator.

What is synthetic division?

A compressed long division for divisors of the form x − c: one row of bring-down, multiply-by-c, add. Dividing x³ − 6x² + 11x − 6 by x − 1 collapses to the single row 1, −5, 6, 0.

How do I enter the problem?

Type the dividend coefficients highest-degree-first (e.g. "1,-6,11,-6") and the value c for the divisor x − c. For x + 2, use c = −2.

How do I read the bottom row?

Every entry except the last is a quotient coefficient; the final entry is the remainder — which the Remainder Theorem identifies as P(c).

What is the Remainder Theorem?

Dividing P(x) by x − c leaves remainder exactly P(c). This page evaluates P(c) directly alongside the tableau so you can watch both agree.

What is the Factor Theorem?

The corollary: x − c is a factor exactly when the synthetic remainder is 0. A zero amber cell certifies c as a root.

Why is the divisor restricted to x − c?

The shortcut exploits the monic linear structure — multiply-by-c plus add replaces full polynomial subtraction. General divisors need the long-division page.

How do I handle x + c divisors?

Rewrite as x − (−c): for x + 2 set c = −2. Sign slips here are the classic synthetic error — negate deliberately.

How do I verify the tableau?

Rebuild with divisor × quotient + remainder, and compare the amber remainder against direct P(c). Both checks run automatically here.

Where is synthetic division used?

Root hunting (rational-root test loops), deflation after finding a zero, Horner-speed evaluation, and partial-fraction setup.