Synthetic Division Calculator

Divide a polynomial by a linear divisor and find quotient coefficients and the remainder instantly.

Polynomial synthetic division

Enter coefficients from highest to lowest power, including zero coefficients for missing powers.

Polynomial coefficients

Separate coefficients with commas and include zero for every missing power.

Divisor value c

For a divisor written as x minus c, enter c with its displayed sign.

About synthetic division

Synthetic division is a compact method for dividing a polynomial by a linear expression whose leading coefficient is one. Instead of writing variables and powers throughout long division, the method works only with the ordered coefficients. For a divisor described as x minus c, the number c drives each multiplication and addition. The final row contains the quotient coefficients followed by the remainder. The process starts by bringing down the leading coefficient unchanged. Multiply that number by c and add the product to the next coefficient. Multiply the new total by c, add it to the following coefficient, and continue until every coefficient has been used. All entries except the last describe a polynomial whose degree is one less than the original. The last entry is the remainder. This calculator applies those Horner-style steps using the numbers provided and displays results rounded to six decimal places. Coefficient order is crucial. Enter the coefficient of the highest power first and move downward one power at a time until the constant term. A missing power must be represented by zero. For example, a cubic containing no squared term still needs four entries, with zero in the second position. Omitting that zero shifts every later coefficient to the wrong power and changes the polynomial being divided. Synthetic division connects directly to the remainder theorem. When a polynomial is divided by x minus c, the remainder equals the polynomial evaluated at c. A remainder of zero therefore means c is a root and x minus c is an exact factor. This makes the method useful for testing possible rational roots, factoring higher-degree expressions, simplifying rational functions, and evaluating polynomials efficiently. It is frequently taught alongside the factor theorem and polynomial long division. The calculator supports real positive, negative, zero, and decimal coefficients. It expects a monic linear divisor represented by c rather than a full expression. If the divisor is x plus 2, rewrite it as x minus negative 2 and enter negative 2. A divisor with a leading coefficient other than one should first be factored or handled with polynomial long division. The output gives coefficients rather than constructing a formatted symbolic expression, avoiding ambiguity about variable names and making it easy to copy the result into another calculation.

Synthetic division examples

Coefficients and divisor value	Quotient and remainder	Interpretation
1, -6, 11, -6 with c equal to 1	Quotient 1, -5, 6; remainder 0	The divisor is an exact factor.
2, 3, -5 with c equal to 2	Quotient 2, 7; remainder 9	Evaluating the polynomial at 2 also gives 9.
1, 0, -4 with c equal to 2	Quotient 1, 2; remainder 0	The zero preserves the missing linear term.
1, 5, 6 with c equal to negative 2	Quotient 1, 3; remainder 0	Entering negative 2 represents division by x plus 2.

How to use synthetic division

Enter polynomial coefficients from the highest power through the constant term.
Insert a zero coefficient wherever a power is missing.
Enter c from the divisor x minus c.
Select Divide Polynomial and read the quotient coefficients and remainder.

Synthetic division FAQ

When can synthetic division be used?

The standard method applies when dividing by a monic linear expression written as x minus c. More general divisors require an extended method or polynomial long division.

Why must I enter zero coefficients?

Each position corresponds to a specific descending power of the variable. A zero holds the place of a missing power so later terms remain aligned correctly.

What does a zero remainder mean?

It means the division is exact and the chosen linear divisor is a factor. Equivalently, c is a root of the original polynomial.

How do I enter a divisor written as x plus a number?

Rewrite the plus expression as subtraction of a negative number. For x plus 2, enter negative 2 as the divisor value c.

How is synthetic division related to Horner's method?

Both repeatedly multiply by one value and add the next coefficient. Horner's method evaluates a polynomial, while synthetic division retains the intermediate values as quotient coefficients.