How to Use Synthetic Division to Divide Polynomials

A step-by-step guide to doing synthetic division on any polynomial

Reviewed by Grace Imson, MA

Last Updated: March 28, 2024 References

Synthetic division is a shorthand method of dividing polynomials where you divide the coefficients of the polynomials, removing the variables and exponents. It allows you to add throughout the process instead of subtract, as you would do in traditional long division.^{[1]
X
Research source} If you want to know how to divide polynomials using synthetic division, just follow these steps.

Things You Should Know

Reverse the sign of the constant in the divisor and write it in a box. Then, write the coefficients in the dividend in a line next to it.
Bring down the first coefficient, multiply it by the number in the box, and write the result under the second coefficient. Then, add the result and the second coefficient.
Repeat the steps for the remaining coefficients. Remove 1 power from the dividend variables, write them next to the new coefficients, and add the remainder.

Steps

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For this example, you will be dividing x³ + 2x² - 4x + 8 by x + 2. Write the first polynomial equation, the dividend, in the numerator and write the second equation, the divisor, in the denominator.
The constant in the divisor, x + 2, is positive 2, so reversing the sign of the constant would give you -2.

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If the opposite of the constant term is -2, we put the -2 inside a box. A common mistake is to put a 2 inside the box, watch out here.
^{[2]
X
Research source} Write the terms from left to right, just as they appear. It should look like this: -2| 1 2 -4 8.
5
Bring down the first coefficient. Bring down the first coefficient, 1, below itself. It should look like this:
- -2| 1  2  -4  8
      ↓
      1
6
Multiply the first coefficient by the divisor and place it under the second coefficient.^{[3]
X
Research source} Simply multiply 1 by -2 to get -2 and write this product under the second term, 2. Here's how it would look:
- -2| 1  2  -4  8
          -2
      1
7
Add the second coefficient and the product and write the answer below the product. Now take the second coefficient, 2, and add it to -2. The result is 0. Write this result below the two numbers, just as you would in long division. Here's how it would look:
- -2| 1  2  -4  8
          -2
      1   0
8
Multiply this sum by the divisor and place the result under the third coefficient. Now take the sum, 0, and multiply it by the divisor, -2. The result is 0. Place this number below 4, the third coefficient. It should look like this:
- -2| 1  2  -4  8
          -2  0
      1
9
Add the product and the third coefficient and write the result under the product. Add 0 and -4 to get -4 and write this answer below the 0. Here's how it would look:
- -2| 1  2  -4  8
          -2   0
      1   0   -4
10
Multiply this number by the divisor, write it under the last coefficient, and add it to the coefficient. Now, multiply -4 by -2 to get 8, write this answer under the fourth coefficient, 8, and add this answer to the fourth coefficient. 8 + 8 = 16, so this is your remainder. Write this number below the product. Here's how it would look:
- -2| 1  2  -4  8
          -2   0   8
      1   0   -4   |16
11
Place each of the new coefficients next to a variable of one less power than their original corresponding variables. In this case, the first sum, 1, is placed next to an x to the second power (one less than three). The second sum, 0, is placed next to an x, but the result is zero, so you can remove this term. And the third coefficient, -4, becomes a constant, a number without a variable, since the original variable was x. You can write an R next to the 16, because that is the remainder. Here's how it would look:
- -2| 1  2  -4  8
          -2   0   8
      1   0   -4   |16
      x²   + 0x    - 4    R 16
  
  x² - 4 R16
The final answer is the new polynomial, x² - 4, plus the remainder, 16, over the original divisor, x + 2. Here's how it would look: x² - 4 +16/(x +2).

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How do I solve synthetic division if the divisor is more than three terms?

Community Answer

The divisor cannot be more than three terms. In synthetic division, the divisor is always in the form x-a.

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What is the benefit of synthetic division?

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You can do it on a smaller piece of paper. Seriously, that's it. Synthetic is computationally identical to long division, but instead of writing powers of x everywhere, it uses its layout to keep the exponents straight.

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How do I solve a problem dividing a polynomial by 2x-1 in synthetic division?

Community Answer

2x-1 is actually the same as x-(1/2), so you can do synthetic division putting +(1/2) in the box.

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Tips

To check your answer, multiply the quotient by the divisor and add the remainder. It should be the same as the original polynomial.^{[4]
X
Research source}

(divisor)(quotient)+(remainder)

(x + 2)(x² - 4) + 16

Using FOIL method, multiply.

(x³ - 4x + 2x² - 8) + 16

x³ + 2x² - 4x - 8 + 16

x³ + 2x² - 4x + 8

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