Skip to content

Synthetic Division Calculator

MathAlgebra Tools

Write 0 for every missing power. x³ − 12x² − 42 is 1, -12, 0, -42.

For x + 2, c is -2.

Quotient —
Remainder
—
Dividend
—
Divisor
—
Is the divisor a factor?
—

Synthetic division divides a polynomial by a linear factor x − c using only its coefficients: bring down the first, multiply by c, add to the next, repeat. For x³ − 12x² − 42 divided by x − 3, the bottom row is 1, −9, −27, −123, so the quotient is x² − 9x − 27 and the remainder is −123.

About this tool

Synthetic division is the shortcut algebra students learn for dividing a polynomial by x − c, and it is the fastest way to test a candidate root found with the rational root theorem. Type the dividend's coefficients from the highest power down and the value of c. The calculator returns the quotient polynomial, the remainder, a yes-or-no on whether x − c is a factor, and the full synthetic division table: each coefficient, the carried value multiplied by c, and the running sum on the bottom row. The working is also written out line by line so you can compare it with a handwritten version. One limit: the method only handles linear divisors with leading coefficient 1. To divide by 2x − 3, divide by x − 1.5 and then halve every quotient coefficient; the remainder stays the same.

How to use it

  1. Enter the dividend coefficients

    List them from the highest power down, separated by commas or spaces. Put 0 in place of any missing power, or the columns shift.

  2. Enter c

    For a divisor x − 4 type 4; for x + 4 type -4. Decimals are accepted.

  3. Read the quotient and remainder

    The quotient has degree one less than the dividend. A remainder of 0 means x − c divides the polynomial exactly.

  4. Check the table and steps

    Each table row is one column of the synthetic division layout, and the steps show every multiply-and-add.

Examples

(x³ − 12x² − 42) ÷ (x − 3)

Dividend coefficients (highest power first)
1, -12, 0, -42
Divisor root c, dividing by (x − c)
3

Result Quotient: x^2 – 9x – 27
Remainder: -123
Dividend: x^3 – 12x^2 – 42
Divisor: x – 3
Is the divisor a factor?: No, P(3) = -123

  1. Bring down 1
  2. 1 × 3 = 3; -12 + 3 = -9
  3. -9 × 3 = -27; 0 + -27 = -27
  4. -27 × 3 = -81; -42 + -81 = -123
  5. Quotient coefficients: 1, -9, -27; remainder -123

The missing x term is entered as 0. The bottom row reads 1, -9, -27, -123, so the quotient is x² − 9x − 27 with remainder -123.

(x³ − 6x² + 11x − 6) ÷ (x − 1)

Dividend coefficients (highest power first)
1 -6 11 -6
Divisor root c, dividing by (x − c)
1

Result Quotient: x^2 – 5x + 6
Remainder: 0
Dividend: x^3 – 6x^2 + 11x – 6
Divisor: x – 1
Is the divisor a factor?: Yes, (x – 1) is a factor

  1. Bring down 1
  2. 1 × 1 = 1; -6 + 1 = -5
  3. -5 × 1 = -5; 11 + -5 = 6
  4. 6 × 1 = 6; -6 + 6 = 0
  5. Quotient coefficients: 1, -5, 6; remainder 0

The remainder is 0, so x − 1 is a factor and the cubic equals (x − 1)(x² − 5x + 6) = (x − 1)(x − 2)(x − 3).

(2x² + 3x − 4) ÷ (x + 2)

Dividend coefficients (highest power first)
2, 3, -4
Divisor root c, dividing by (x − c)
-2

Result Quotient: 2x – 1
Remainder: -2
Dividend: 2x^2 + 3x – 4
Divisor: x + 2
Is the divisor a factor?: No, P(-2) = -2

  1. Bring down 2
  2. 2 × -2 = -4; 3 + -4 = -1
  3. -1 × -2 = 2; -4 + 2 = -2
  4. Quotient coefficients: 2, -1; remainder -2

Dividing by x + 2 means c = -2. P(-2) = 8 − 6 − 4 = -2, which matches the remainder.

How it is calculated

b₀ = a₀; bₖ = aₖ + c · bₖ₋₁ for k = 1 … n; remainder R = bₙ = P(c)

aₖ
coefficients of the dividend, highest power first
c
the root of the divisor x − c
bₖ
bottom-row sums; b₀ … bₙ₋₁ are the quotient coefficients
R
the remainder, equal to P(c) by the remainder theorem

This is Ruffini's rule, which is also Horner's scheme for evaluating a polynomial. Each bottom-row value is multiplied by c and added to the next coefficient. Because the last sum equals P(c), a zero remainder proves c is a root and x − c is a factor (the factor theorem). The quotient has degree n − 1.

When not to use it

  • Synthetic division only works for linear divisors of the form x − c.
  • Use polynomial long division for x² + 1 or any divisor of degree 2 or more.
  • For a divisor like 3x + 1, the quotient must be rescaled afterwards.

Common mistakes

  • Skipping a zero coefficient: enter x³ − 42 as 1, 0, 0, -42.
  • Getting the sign of c wrong: dividing by x + 5 uses c = -5, not 5.
  • Reading the last bottom-row number as part of the quotient; it is the remainder.

Frequently asked questions

How do I enter a polynomial with missing terms?

Write a 0 for every power that does not appear. For 4x⁴ − 3x + 2 the coefficients are 4, 0, 0, -3, 2. Without the zeros the calculator would read a cubic, and every column would be off by one place.

What does a remainder of zero mean?

It means x − c divides the polynomial exactly, so c is a root. By the factor theorem the polynomial equals (x − c) times the quotient. For x³ − 6x² + 11x − 6 with c = 1 the remainder is 0, giving (x − 1)(x² − 5x + 6).

Why is the remainder equal to P(c)?

Writing P(x) = (x − c)Q(x) + R and substituting x = c makes the first term vanish, leaving P(c) = R. This is the remainder theorem, and it is why synthetic division doubles as a quick way to evaluate a polynomial at a point.

Can I divide by something like 2x − 3?

Yes, with one extra step. Divide by x − 1.5 here, then divide each quotient coefficient by 2. The remainder is unchanged, because 2x − 3 = 2(x − 1.5) and the factor 2 is absorbed into the quotient only.

How is synthetic division different from long division?

It carries out the same arithmetic as long division by x − c but drops the variables and replaces subtraction with addition by using c instead of −c. That makes it shorter and less error-prone, but it only works for linear divisors with leading coefficient 1.

Which values of c are worth testing for a root?

List candidate rational roots p/q, where p divides the constant term and q divides the leading coefficient. Test each as c. Whenever the remainder is 0, you have a root, and the quotient is a lower-degree polynomial you can keep factoring, for example with the quadratic formula once it reaches degree 2.

Cite this tool

Using this tool in an assignment, article or report? Copy a citation, or a link for your website.

APA

Tools Directory Editorial Team. (2026, October 24). Synthetic Division Calculator. Tools Directory. Retrieved October 26, 2026, from https://tools.milunir.com/math/algebra-tools/synthetic-division-calculator/

MLA

Tools Directory Editorial Team. “Synthetic Division Calculator.” Tools Directory, 24 Oct. 2026, https://tools.milunir.com/math/algebra-tools/synthetic-division-calculator/. Accessed 26 Oct. 2026.

Chicago

Tools Directory Editorial Team. “Synthetic Division Calculator.” Tools Directory. Last modified October 24, 2026. Accessed October 26, 2026. https://tools.milunir.com/math/algebra-tools/synthetic-division-calculator/.

Link<a href="https://tools.milunir.com/math/algebra-tools/synthetic-division-calculator/">Synthetic Division Calculator – Tools Directory</a>