What number do I use for x plus a number?

Rewrite the divisor as x minus c. For x plus 3, c is negative 3; for x minus 3, c is positive 3.

Why must I enter zeros for missing terms?

Each position represents a power of x. A zero preserves the correct column for a missing power, such as the x squared term in x cubed plus 2x minus 1.

What does the remainder tell me?

For division by x minus c, the remainder equals P(c). A remainder of zero means x minus c is a factor and c is a root.

Can synthetic division use fractions or decimals?

Yes. This calculator converts finite decimals and fractions to exact reduced fractions rather than rounding them.

Can I divide by a quadratic using synthetic division?

This calculator uses the standard form for monic linear divisors x minus c. Use polynomial long division for quadratic or other polynomial divisors.

Synthetic Division Calculator with Remainder

Divide a polynomial by x − c using exact synthetic division. Your coefficients are processed only in your browser and are not uploaded or stored.

Get the quotient, remainder, synthetic table, step-by-step multiply-and-add work, and an exact check that the remainder equals P(c).

Enter coefficients and c

Synthetic division result

Quotient with remainder

x² + 4/(x − 2)

Remainder: 4

Quotientx²

Remainder4

P(c)P(2) = 4

Identityx³ − 2x² + 4 = (x − 2)(x²) + 4

Synthetic division table

Synthetic division steps

How synthetic division works

1. Set up the coefficients

List coefficients in descending powers. For x³ − 2x² + 4, use 1, −2, 0, 4; the zero holds the missing x column.

2. Bring down

Copy the leading coefficient to the bottom row. It becomes the leading coefficient of the quotient.

3. Multiply and add

Multiply the latest bottom value by c, place the product under the next coefficient, and add. Continue to the final column.

4. Read the result

The last bottom value is the remainder. The preceding values are the quotient coefficients, beginning one degree below the dividend.

Synthetic division recurrence

b₀ = a₀ and bₖ = aₖ + c·bₖ₋₁; quotient coefficients = b₀…bₙ₋₁, remainder = bₙ = P(c)

This shortcut is equivalent to dividing P(x) by the monic linear polynomial x − c. The exact identity is P(x) = (x − c)Q(x) + P(c).

Worked example with a remainder

For P(x) = x³ − 2x² + 4 and divisor x − 2, use coefficients 1, −2, 0, 4 and c = 2. Bring down 1; multiply by 2 and add to −2 to get 0; multiply 0 by 2 and add to 0; then add the final product 0 to 4.

The quotient is x² and the remainder is 4, so x³ − 2x² + 4 = (x − 2)x² + 4. The Remainder Theorem confirms P(2) = 4.

Exact arithmetic, privacy, and limits

The calculator reduces all integers, finite decimals, and fractions with BigInt arithmetic. It does not use floating-point approximations, and calculations stay on this page.

Enter 1 to 31 coefficients (degree 0 through 30), with no more than 30 digits in each numerator or denominator. Empty entries, zero denominators, scientific notation, variables, and mixed-number notation are not accepted. For non-monic or higher-degree divisors, use polynomial long division. Extremely large exact intermediate values are stopped with a clear message. Last reviewed: 4 August 2026.

Synthetic division FAQs

How do I use synthetic division?

Enter coefficients from the highest power to the constant, including zeros for missing terms. Enter c from the divisor x − c, bring down the first coefficient, then multiply by c and add across the row.

What value of c represents x + 3?

Use c = −3 because x + 3 = x − (−3). The sign is reversed from the constant shown in the divisor.

Why do missing terms need zero coefficients?

Each coefficient must stay aligned with its power. For x³ + 2x − 1, enter 1, 0, 2, −1 so the x² column remains present.

What does a zero remainder mean?

It means x − c divides the polynomial exactly. By the Factor Theorem, x − c is a factor and c is a root.

Can I enter fractions and decimals?

Yes. Use values such as 3/4, -2.5, or .125. Finite decimals are converted to reduced fractions for exact calculation.

Can synthetic division divide by a quadratic?

This standard calculator is for divisors of the form x − c. Use the polynomial long division calculator for a quadratic, a higher-degree polynomial, or a non-monic linear divisor.