Education
Synthetic Division Calculator
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.
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
| Step | Next coef | Carry (prev x c) | Sum |
|---|---|---|---|
| 0 | 1 | bring down | 1 |
| 1 | -5 | 1 x 2 = 2 | -3 |
| 2 | 6 | -3 x 2 = -6 | 0 |
The last sum in the rightmost column is the remainder. The earlier sums are the quotient coefficients in order.
Quotient
x - 3
Remainder: 0
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.
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
- Polynomial long division calculator for any polynomial divisor (not just x - c).
- Factoring calculator when you have a candidate root from the rational-roots theorem and want to factor cleanly.
- Remainder calculator for integer division remainders rather than polynomial remainders.
- All education calculators.
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).
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