Synthetic division calculator

A synthetic division calculator divides a polynomial by a linear divisor such as x − 2, using only the coefficients. Our Polynomial Division Calculator does it with every step shown: bring down, multiply, add, repeat. It gives the quotient and the remainder, and says from the factor theorem whether the divisor is a factor.

The calculator does long division and synthetic division, so choose Synthetic to switch it to this method. It opens on long division of 2x³ − 3x² + 5 by x − 2.

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Polynomial Division Calculator

Divide polynomials by long or synthetic division

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How to use the synthetic division calculator

1. In the Division section, choose Synthetic.
2. Type the polynomial you are dividing in Dividend, such as x^3-4x^2+x+6. Press the right arrow key after each power to come back down.
3. Type the divisor in Divisor, such as x-2. It must be degree 1: x − c, or ax − b such as 2x − 1.
4. Read the quotient and the remainder, and the remainder theorem and factor theorem lines underneath.
5. Follow the working, which shows the synthetic division table filling in one column at a time.

If the divisor has degree 2 or more, such as x² + 1, the calculator asks you to choose long division instead, because synthetic division only works for linear divisors. The Examples list includes an exact factor, a division with a remainder, a quadratic divisor and one with fractions. For practice, Practice and help sets remainder questions with worked solutions.

How synthetic division works

1. Set the divisor equal to zero to find c. For x − 2, c = 2. For x + 3, c = −3.
2. Write the coefficients of the dividend in order, with a 0 for any missing power.
3. Bring the first coefficient straight down.
4. Multiply it by c, write the result under the next coefficient, and add.
5. Repeat until you reach the end. The last number is the remainder, and the others are the coefficients of the quotient, one power lower than the dividend.

Example 1: an exact factor

Divide x³ − 4x² + x + 6 by x − 2. Here c = 2 and the coefficients are 1, −4, 1, 6.

The bottom row gives the quotient x² − 2x − 3 and a remainder of 0. Since the remainder is 0, x − 2 is a factor, and the calculator says "(x − 2) is a factor". The quotient factors further, so x³ − 4x² + x + 6 = (x − 2)(x − 3)(x + 1).

Example 2: a remainder and a negative c

Divide 2x³ + 3x² − 5x + 7 by x + 3. Setting x + 3 = 0 gives c = −3.

The quotient is 2x² − 3x + 4 and the remainder is −5, so:

The calculator writes the answer as 2x² − 3x + 4 − 5/(x + 3), and checks the remainder theorem: f(−3) = −5.

Example 3: a missing power

Divide x⁴ − 3x² + 2x − 5 by x − 1. There is no x³ term, so its coefficient is 0: the row is 1, 0, −3, 2, −5. Bringing down and adding with c = 1 gives 1, 1, −2, 0, −5. The quotient is x³ + x² − 2x and the remainder is −5. Leaving out the 0 is the most common way to get synthetic division wrong.

Check you have the method:

Dividing by ax − b

Synthetic division also works when the divisor's first coefficient is not 1, with one extra step. For 2x − 1, use c = 1/2, run the table as normal, then divide the quotient by 2. The remainder stays the same.

Dividing 3x³ − 2x + 4 by 2x − 1 gives the row 3, 3/2, −5/4 and a remainder of 27/8. Dividing the quotient by 2 gives (3/2)x² + (3/4)x − 5/8. The calculator does this step and says "The divisor's leading coefficient is 2, so divide the quotient by it".

The remainder theorem and the factor theorem

The remainder theorem says that when a polynomial f(x) is divided by x − c, the remainder is f(c). So in Example 2, f(−3) = 2(−27) + 3(9) − 5(−3) + 7 = −5, the same as the table. The factor theorem follows: x − c is a factor exactly when f(c) = 0.

This is how to find factors of a cubic. Try small values of c until f(c) = 0, then use synthetic division to get the quadratic that is left. The Factoring Calculator does this search for you with the rational root theorem, and our factoring calculator guide walks through it.

Synthetic division or long division?

Choose Long division in the calculator for a quadratic or higher divisor. Our polynomial guide covers the terms used here.

Common mistakes

• Using the wrong sign for c. For x + 3, c is −3, not 3. Set the divisor equal to zero.
• Missing a zero coefficient. Every power from the highest down needs a place, so x⁴ − 3x² needs 0 for x³ and for x.
• Subtracting instead of adding. Synthetic division always adds. The sign change is already built into c.
• Getting the quotient's degree wrong. It is one less than the dividend's. A cubic divided by x − c gives a quadratic.
• Forgetting to divide by a. With a divisor like 2x − 1, divide the quotient by 2 at the end.

Practise polynomial division in a short timed paper

The factor and remainder theorems come up in our A-level Maths: Algebra & Functions daily papers, which open as AP Precalculus in the US. Each daily paper is a short timed paper of 8 questions worth 25 to 35 marks, and the papers mix topics, so division will not come up in every one.

Read next

Maths · AlgebraRemainder theorem: statement, proof and worked examplesThe remainder theorem explained: why f(a) is the remainder, dividing by ax − b, finding unknown coefficients, and its link to the factor theorem.