Synthetic division calculator

What is synthetic division?

A shortcut for dividing a polynomial by a linear factor x − c using only the coefficients. Bring down the first coefficient, multiply by c, add to the next coefficient, and repeat.

Example

Divide 5x² − 3x − 36 by x − 3: bring down 5; 5 × 3 = 15, −3 + 15 = 12; 12 × 3 = 36, −36 + 36 = 0. The quotient is 5x + 12 with remainder 0 (OpenStax College Algebra).

How do I enter missing powers?

Type a 0 for each missing term. For −9x⁴ + 10x³ + 7x² − 6, enter −9, 10, 7, 0, −6.

What if I divide by x + 2?

Write it as x − (−2) and enter c = −2. Dividing 4x³ + 10x² − 6x − 20 by x + 2 gives 4x² + 2x − 10 with remainder 0.

What does the remainder tell me?

By the remainder theorem it equals the polynomial’s value at x = c. A remainder of 0 means x − c is a factor and c is a root.

How do I find all the roots of a cubic with synthetic division?

Use the rational root theorem to list candidates, which are plus or minus a factor of the constant term over a factor of the leading coefficient, and test them one at a time. For x³ − 7x + 6 (enter 1, 0, −7, 6) try c = 1: the remainder is 0, so x − 1 is a factor and the quotient is x² + x − 6. Dividing that by x − 2 leaves remainder 0, and the last factor is x + 3, so the roots are 1, 2 and −3. Trying c = −3 on the original also gives remainder 0.

Can I divide by 2x − 1 or another divisor with a leading coefficient?

Not directly. Rewrite the divisor as 2(x − 1/2), divide by x − 1/2 with c = 0.5, then divide every coefficient of the quotient by 2. The remainder is unchanged. The method itself only handles divisors of the form x − c.

How the steps work

Write the coefficients in order of descending power, with a 0 for every missing power. Bring down the first coefficient. Multiply it by c, write the product under the next coefficient and add. Repeat across the row. The last number is the remainder and the numbers before it are the coefficients of the quotient, whose degree is one less than the original polynomial. The calculator shows the three rows of the standard layout: the coefficients, the products of c, and the sums. The method is just long division written without the powers of x, which is why it works only when the divisor is x − c with leading coefficient 1.

Worked example: remainder when the division is not exact

Divide 2x³ − 6x² + 2x − 1 by x − 3, so c = 3. Bring down 2. Then 2 × 3 = 6 and −6 + 6 = 0. Next 0 × 3 = 0 and 2 + 0 = 2. Finally 2 × 3 = 6 and −1 + 6 = 5. The quotient is 2x² + 2 (the x term has coefficient 0) and the remainder is 5, so 2x³ − 6x² + 2x − 1 = (x − 3)(2x² + 2) + 5. Checking with the remainder theorem: the polynomial evaluated at x = 3 is 2(27) − 6(9) + 2(3) − 1 = 5, the same number. The remainder is not zero, so x − 3 is not a factor.

Common mistakes

Leaving out a zero for a missing power shifts all the coefficients and gives a wrong answer; x³ − 7x + 6 must be entered as 1, 0, −7, 6. The sign of c is the second trap: for the divisor x + 2, c is −2, because x + 2 = x − (−2). The third is dividing by a non-linear divisor such as x² + 1, which synthetic division does not handle. Fourth, a nonzero remainder does not mean you made an error; it means the divisor is not a factor, and its value is the polynomial's value at c. Finally, the quotient is only exact when the remainder is zero; otherwise the full answer includes the fraction remainder ÷ (x − c).

Sources: OpenStax College Algebra, section 5.4 (dividing polynomials: synthetic division) and section 5.5 (zeros of polynomial functions: remainder theorem, factor theorem, rational zero theorem). Related: polynomial calculator, factor calculator and quadratic formula calculator.