2026-09-29 9 min

Polynomial Remainder Calculator

Find the remainder of P(x) ÷ (x − c) with synthetic division and verify it equals P(c). Free online remainder theorem calculator.

Calculator

Remainder Inputs

Coefficients (highest degree first)

Comma-separated; include 0 for missing powers. Results update automatically.

Value c in (x − c) Help

Divisor constant

Results update live as you type

Status:P(x) = x^3 - 6x^2 + 11x - 6R = 0Exact factor

Remainder Summary

Remainder R0

P(c) check0

Quotientx^2 - 5x + 6

Synthetic Tableau (c = 1)

Coeffs: 1 | -6 | 11 | -6

Step 1: 1 + (1 × 1) = -5

Step 2: -6 + (1 × -5) = 6

Step 3: 11 + (1 × 6) = 0

→ Quotient: x^2 - 5x + 6 · Remainder: 0

Results

Optimal

Remainder R

Remainder of P(x) ÷ (x − c)

Check P(c)

Direct evaluation — must equal R

Quotient Q(x)

x^2 - 5x + 6

One degree lower than P(x)

Degree of P

Highest power with nonzero coefficient

(x − c) a Factor?

Yes — P(c) = 0

Factor theorem verdict

Synthetic tableau arithmetic is exact in double precision and every remainder is cross-checked against direct P(c) evaluation.

Visualizations

Synthetic Tableau Running Values

1b0

5b1

6b2

Division Completeness

Done100%

|Remainder| vs |P(c)|

|R|0

Formula

Dividing by (x − c) leaves a constant remainder. The remainder theorem says that constant is exactly P(c).

P(x) = (x − c)·Q(x) + R, with R = P(c)

Alternative Form

Synthetic step: b_k = a_k + c·b_{k+1}, remainder = last b

Variables

Symbol	Name	Unit	Description
P(x)	Dividend polynomial	expression	Polynomial being divided
c	Divisor constant	number	Divisor is (x − c)
Q(x)	Quotient	expression	Result polynomial, one degree lower
R	Remainder	number	Constant equal to P(c)

Explanation

The tableau folds each coefficient through c; the final fold has nowhere to go, so it becomes the remainder.

Derivation

P(x) − P(c) has root x = c, so P(x) − P(c) = (x − c)·Q(x); setting x gives R = P(c). Synthetic division is Horner evaluation in disguise.

Step-by-Step Calculation

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

Step 1

Write P(x) and the divisor (x − c).

P(x) = (x − c)·Q(x) + R

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

divisor:x − (1)

Result: P(x) = x^3 - 6x^2 + 11x - 6, c = 1

Step 2

Bring down the leading coefficient.

b_n = a_n

leading:1

Intermediate: col0: 1; col1: -6; col2: 11; col3: -6

Result: First tableau value = 1

Step 3

Multiply by c and add across the row.

b_k = a_k + c·b_{k+1}

c:1

Intermediate: -6 + 1 = -5; 11 + -5 = 6; -6 + 6 = 0

Result: Tableau: 1, -5, 6, 0

Step 4

Read off the quotient and remainder.

R = last value, Q = the rest

Q:x^2 - 5x + 6

R:0

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

Step 5

Verify with direct evaluation P(c).

P(c) should equal R

Pc:0

R:0

Result: P(1) = 0 ✓ matches R

Step 6

Apply the factor theorem.

R = 0 ⟺ (x − c) is a factor

verdict:factor

Result: (x − 1) divides P(x) exactly

Final Answer

Worked Examples

See how the calculator works with different scenarios.

Scenario

Testing whether x = 1 is a root.

Inputs

coeffs:1, -6, 11, -6

c:1

Calculation

Tableau: 1, −5, 6, 0 → R = 0

Result

R = 0; (x − 1) is a factor

Interpretation

Since P(1) = 0, x = 1 is a root of the cubic.

Understanding Your Result

Your Result: Remainder R = P(c) with quotient and factor verdict

Typical range: R = 0 means an exact factor; any other value is the leftover constant.

What This Means

The remainder is the height of the polynomial at x = c — what is left over after dividing by (x − c).

Why It Matters

It tests roots, factors polynomials, evaluates efficiently, and checks division work in one short tableau.

Professional Interpretation

Read R = 0 as "c is a root"; read R ≠ 0 as "P(c) is the correction term". Always cross-check with direct evaluation.

Recommendations

•Include 0 for missing powers
•Verify R against direct P(c)
•Read quotient as one degree lower
•Use R = 0 to claim factors only
•Watch signs when c is negative

Educational Content

Introduction

This calculator divides any polynomial by (x − c) using synthetic division and proves the answer by evaluating P(c).

Concept Explanation

Each synthetic step folds a coefficient through c; the unfolding at the end leaves exactly P(c) as the remainder.

Applications

▸Testing candidate roots

▸Factoring cubics and quartics

▸Fast polynomial evaluation

▸Checking long division

▸Control-system pole checks

▸CRC-style remainder codes

Benefits

•Full synthetic tableau shown
•Automatic P(c) cross-check
•Quotient polynomial included
•Factor verdict built in
•Sign-safe arithmetic

Limitations

•Divisor must be (x − c)
•Single-variable polynomials
•Display rounds to 6 decimals

Best Practices

•List coefficients highest-degree first
•Pad missing powers with zero
•Multiply-then-add left to right
•Confirm with P(c)
•Keep c signs explicit

Common Misconceptions

Myth: The remainder is a polynomial.

Truth: For divisor (x − c) the remainder is always a constant, P(c).

Myth: You can skip zero coefficients.

Truth: Every missing power needs a 0 placeholder or the tableau shifts.

Myth: A nonzero remainder means a mistake.

Truth: Nonzero remainders are normal — they equal P(c).

Myth: Synthetic division works for any divisor.

Truth: Plain synthetic division needs (x − c); higher-degree divisors need long division.

Practical Tips

•Pad missing powers with 0 first
•Bring down, multiply, add — repeat
•Verify with direct P(c)
•Negative c flips multiply signs
•Quotient is one degree lower
•R = 0 certifies a factor
•Re-run after any edit

Common Mistakes

Omitting zero coefficients

Write a 0 for every missing power before starting.

Adding before multiplying

Each column is multiply-then-add, strictly left to right.

Sign errors with negative c

Multiply carefully: a negative c negates each carried value.

Calling the quotient the remainder

The remainder is only the last tableau value.

Claiming a factor from R ≠ 0

Only R = 0 proves (x − c) is a factor.

Assumptions

This calculator makes the following assumptions to provide estimates:

Monic linear divisor: The divisor has the form (x − c).

Highest-degree-first order: Coefficients are listed from leading term to constant.

Real arithmetic: Coefficients and c are real numbers.

Exact coefficients: Inputs are treated as exact values.

Single variable: P is a polynomial in one variable x.

Limitations

Be aware of these limitations when using this calculator:

Linear divisors only: Quadratic or higher divisors need polynomial long division.

No symbolic output: Work is numeric, not letter-based algebra.

Display rounding: Long decimals show 6 digits.

One divisor at a time: Each run tests a single value of c.

Frequently Asked Questions

Answers to common questions about this calculator.

What is the polynomial remainder theorem?

When a polynomial P(x) is divided by (x − c), the remainder equals P(c) — the polynomial evaluated at c. So you can find the remainder without doing long division.

How do I use synthetic division to find the remainder?

Write the coefficients, bring down the leading one, then repeat: multiply by c, add to the next coefficient. The last value is the remainder; the earlier values are the quotient coefficients.

What does a zero remainder mean?

A remainder of 0 means (x − c) is a factor of P(x) and c is a root: P(c) = 0. This is the factor theorem, the converse of the remainder theorem.

What is the difference between remainder and quotient?

Dividing P(x) by (x − c) gives P(x) = (x − c)·Q(x) + R. Q(x) is the quotient (one degree lower), R is the constant remainder equal to P(c).

Can c be negative or fractional?

Yes. Synthetic division works for any real c, including negatives and fractions. Just be careful with signs when multiplying and adding.

How do I check my synthetic division?

Evaluate P(c) directly with the original polynomial. If it matches the synthetic remainder, your tableau is correct — this page does that check automatically.

Does the remainder theorem work for divisors like (2x − 1)?

Directly it applies to monic divisors (x − c). For (ax − b), factor out a first: divide by a·(x − b/a), so the remainder is still P(b/a).

Where is the remainder theorem used?

Testing candidate roots, factoring polynomials, evaluating polynomials quickly (Horner form), error-checking division, and in coding theory (e.g. CRC checks).

What if the polynomial is missing a power of x?

Insert a 0 coefficient for every missing power. For x³ + 2x + 1 use coefficients 1, 0, 2, 1 — skipping the zero breaks the tableau.