Synthetic division is a shortcut for polynomial division when the divisor is of the form x – a. Only numeric coefficients of the dividend are used when dividing with synthetic division.
Example 1
Divide (2 x – 11 + 3 x 3) by ( x – 3).
First, this problem is done in the traditional manner. Then it is done by using the synthetic division method.
In the traditional manner, 
The answer is 
.
To do the problem using synthetic division, follow this procedure:
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Write the polynomial being divided in descending order. Then write only its coefficients and constant, using 0 for any missing terms.
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Write the constant, a, of the divisor, x – a, to the left. In this problem, a = 3 because you use the additive inverse of the constant. (Remember, the additive inverse of –3 is 3.)
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Bring down the first coefficient as shown.
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Multiply the first coefficient by the divisor, 3. Then write this product under the second coefficient.
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Add the second coefficient with the product and write the sum as shown.
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Continue this process of multiplying and adding until there is a sum for the last column.
The numbers along the bottom row are the coefficients of the quotient with the powers of x in descending order. The last coefficient is the remainder. The first power is one less than the highest power of the polynomial that was being divided.
The division answer is 
.
Example 2
Divide (5 x 4 + 6 x 3 – 9 x 2 – 7 x + 6) by ( x + 2) using the synthetic method.
To put the divisor, x + 2, into the form x – a, use the constant's negative. That means using x – (+2), so a = –2.
The answer is 
.