Skip to main content
Calculator Answers

Synthetic Division Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

A synthetic division calculator is an algebraic tool that performs division on a polynomial by a linear binomial divisor of the form x minus c. It streamlines polynomial division by utilizing only the coefficients of the terms, avoiding variable writing. The calculator displays the synthetic division table, showing the addition and multiplication steps, and outputs the final quotient polynomial alongside the remainder. High school and college algebra students use this calculator to quickly factor polynomials, find roots, and evaluate functions.

Pick a polynomial, enter its coefficients in descending order, and the calculator divides it by x - c using synthetic division. The result panel shows the quotient and remainder; the synthetic table shows every step.

See also: Polynomial Long Division Calculator, Long Division Calculator, and Unit Circle Calculator.

Quick Answer

Divide polynomials using synthetic division. Enter the numerator polynomial and linear divisor to find the quotient and remainder with step-by-step math.

Polynomial and divisor

Enter coefficients of the dividend in descending order, including zeros for missing terms. Synthetic division supports a divisor of the form x - c (or x + c, which is c = -value).

Comma or space separated, highest power first. · e.g. 1, -5, 6 (for x^2 - 5x + 6)

To divide by x + 3, enter c = -3. · e.g. 2 (for x - 2)

Synthetic table

StepNext coefCarry (prev x c)Sum
01bring down1
1-51 x 2 = 2-3
26-3 x 2 = -60

The last sum in the rightmost column is the remainder. The earlier sums are the quotient coefficients in order.

Result

Quotient

x - 3

Remainder: 0

Dividendx^2 - 5x + 6
Divisorx - (2)
Quotient coefs1, -3
Remainder0

Synthetic division is a compact shortcut that works only when the divisor is x - c. For divisors of higher degree (like x^2 + 1), use polynomial long division instead.

Was this helpful?

Examples

(x^2 - 5x + 6) / (x - 2)

Quotient x - 3, Remainder 0

(2x^3 + 3x^2 - 4x + 5) / (x - 1)

Quotient 2x^2 + 5x + 1, Remainder 6

(x^3 - 8) / (x - 2)

Quotient x^2 + 2x + 4, Remainder 0

(x^2 + 1) / (x + 1)

Quotient x - 1, Remainder 2

How it works

Synthetic division processes the coefficients left to right. Bring down the leading coefficient, multiply by c, add to the next coefficient, and repeat. The last value is the remainder; everything before it is a quotient coefficient.

Step · row[i] = coef[i] + row[i-1] * c

Quotient degree = dividend degree - 1. Remainder is a single number (because the divisor is linear).

Related algebra calculators

Related Calculators

More tools from Education

Polynomial Long Division CalculatorDivide one polynomial by another using polynomial long division. Enter coefficients for the dividend and divisor; the calculator returns the quotient, the remainder, and a step-by-step trace.Long Division CalculatorDivide two whole numbers and see the quotient, remainder, decimal result, and mixed-number form. Handles decimal inputs as a fall-through decimal answer.Unit Circle CalculatorEnter an angle in degrees or radians and the calculator returns the reference angle, quadrant, sin, cos, tan, and the unit-circle coordinate (x, y). Exact values for common angles, decimal otherwise.Matrix CalculatorRun matrix operations on 2x2 and 3x3 matrices: addition, subtraction, scalar multiplication, matrix multiplication, transpose, determinant, and inverse. Not a system solver (see RREF and system of equations).

Frequently asked questions

Synthetic division is a compact shortcut for dividing a polynomial by a linear divisor of the form x - c. Instead of writing out every term, you record only the coefficients in a table and use the value c to fold each step into the next.

Only when the divisor is of the form x - c (a linear factor with leading coefficient 1). For divisors of higher degree, or for divisors like 2x - 5 with a non-unit leading coefficient, use polynomial long division instead.

List the coefficients in descending order of power, separated by commas. Include zeros for any missing terms. For example, x^3 - 2x + 5 has coefficients 1, 0, -2, 5 (the x^2 term is zero).

Enter the value c, not the entire 'x - c' expression. To divide by x - 2, enter c = 2. To divide by x + 3, enter c = -3 (because x + 3 is the same as x - (-3)).

The numbers along the bottom of the synthetic table are the quotient coefficients (read in descending power, starting one degree below the dividend) followed by the remainder as the last entry.

A zero remainder means x - c is an exact factor of the polynomial. This is the basis of the factor theorem: P(c) = 0 if and only if (x - c) divides P(x).