Synthetic Division Calculator
Enter your dividend polynomial and the root r of the divisor (x - r) to perform synthetic division instantly. Get the quotient polynomial, remainder, full division equation, and a step-by-step tableau. Supports decimal and fraction coefficients, non-monic divisors, and synthetic substitution.
Formula
Worked example
Divide x^3 - 6x^2 + 11x - 6 by (x - 1): coefficients 1, -6, 11, -6, root r = 1. Bring down 1. Then: 1*1 + (-6) = -5; 1*(-5) + 11 = 6; 1*6 + (-6) = 0. Quotient coefficients are 1, -5, 6, so Q(x) = x^2 - 5x + 6, and the remainder is 0. Since R = 0, x = 1 is a root, and x^2 - 5x + 6 = (x-2)(x-3) gives roots 1, 2, 3.
How synthetic division works
Synthetic division is a compact shortcut for dividing a polynomial P(x) by a linear factor (x - r). You list the coefficients in descending order of power, placing a zero for any missing term. Bring down the leading coefficient, then for each subsequent column: multiply the running total by r and add the next coefficient. The bottom row gives the quotient coefficients (all but the last) and the remainder (the last number). The entire process is a single row of arithmetic, far faster than polynomial long division.
Polynomial notation vs. coefficient mode
This calculator accepts polynomials two ways. Coefficient mode is fastest: just enter the numbers in descending order, separated by commas, with zeros for missing terms (e.g. x^3 + 2 becomes 1, 0, 0, 2). Polynomial notation lets you type expressions directly in standard algebraic form such as 3x^3 - 2x + 5. The parser handles missing terms automatically. Both modes support decimal and fractional coefficients. For non-monic divisors like (2x - 3), enable the "Non-monic divisor" toggle, set r = 3/2 = 1.5, and set a = 2. The calculator divides by (x - 1.5) and then adjusts the quotient coefficients by dividing by 2, giving the correct result for the original divisor.
The Remainder Theorem and Factor Theorem
The last number in the synthetic division tableau is not just a leftover: by the Remainder Theorem it equals P(r), the value of the polynomial at x = r. This makes synthetic division one of the quickest ways to evaluate a polynomial at a point, sometimes called synthetic substitution. When the remainder is exactly zero, the Factor Theorem tells you (x - r) divides the polynomial evenly, r is a root, and you can factor the quotient further to find the remaining roots. Repeatedly applying this process is the standard method for completely factoring a high-degree polynomial.
Rational Root Theorem and finding roots to try
For a polynomial with integer coefficients, the Rational Root Theorem says every rational root must have the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. Enable the "Show rational root candidates" toggle to see all such candidates. Test each one with synthetic division: a remainder of zero confirms that candidate is a root, and the quotient gives a polynomial of lower degree to work on next. This systematic approach will always find all rational roots of a polynomial with integer coefficients.
Reading the full division equation
Beyond the quotient and remainder, this calculator shows the complete algebraic identity: P(x) = (x - r) * Q(x) + R. This is the formal statement of what synthetic division computes, and it holds for every value of x. Substituting x = r into both sides confirms R = P(r), the Remainder Theorem. The full equation is especially useful when you need to write the division result in a textbook format or verify your work by expanding the right-hand side.
Synthetic division tableau example: x^3 - 6x^2 + 11x - 6 divided by (x - 1)
| x^3 | x^2 | x^1 | x^0 (remainder) | |
|---|---|---|---|---|
| Coefficients | 1 | -6 | 11 | -6 |
| Carry row (x r) | 1 x 1 = 1 | 1 x (-5) = -5 | 1 x 6 = 6 | |
| Bottom row | 1 | -5 | 6 | 0 |
| Meaning | x^2 coef | x^1 coef | x^0 coef | Remainder |
r = 1. Each bottom-row entry = coefficient above + r x entry to its left. Last entry is the remainder.
Frequently asked questions
Why do I need to include zeros for missing terms?
Synthetic division lines up each column with a specific power of x. If a term like x^2 is missing, you must enter 0 in that slot so every power is represented. Otherwise the columns shift and every subsequent number in the bottom row is wrong. In polynomial notation mode this calculator adds the zeros automatically.
Can I divide by a quadratic or higher-degree polynomial?
Standard synthetic division only works for monic linear divisors of the form (x - r). To divide by a non-monic linear like (2x - 3), use the non-monic toggle and set r = 1.5 with a = 2. For divisors of degree 2 or higher you need polynomial long division instead.
What does a remainder of zero mean?
A zero remainder means (x - r) divides the polynomial exactly, so r is a root and (x - r) is a factor. You can then apply synthetic division to the quotient to find the remaining roots, or use the quadratic formula if the quotient is degree 2.
How do I handle a non-monic divisor like (2x - 3)?
Enable the "Non-monic divisor" toggle. Set r to the root of the divisor (for 2x - 3, the root is 3/2 = 1.5) and set a = 2 (the leading coefficient). The calculator performs synthetic division using r = 1.5, then divides every quotient coefficient by a = 2 to account for the factor. The remainder is unchanged.
What is synthetic substitution?
Synthetic substitution uses the same row of arithmetic to evaluate P(r) efficiently. Instead of plugging r into every term of the polynomial, you run the synthetic division algorithm and read the remainder. By the Remainder Theorem, the remainder always equals P(r), making synthetic division one of the fastest evaluation methods for high-degree polynomials.
How do I use the Rational Root Theorem with this calculator?
Enable the "Show rational root candidates" toggle. The calculator lists all p/q values where p divides the constant term and q divides the leading coefficient. Enter each candidate as r and check whether the remainder is zero. This systematically finds all rational roots. Candidates are only shown for polynomials with integer coefficients.