Synthetic Division of a Polynomial — Steps and Examples

Synthetic division is a shortcut for dividing a polynomial by a linear factor of the form x−c, using only the coefficients instead of the full variables. This article defines the method, walks through six worked examples including a wrong-path check, explains why it works through the Remainder Theorem, and names the mistakes that trip students up.

What Synthetic Division Is

Synthetic division is a compact way to divide a polynomial by a linear factor x−c, working only with the coefficients of the dividend and the single number c. It returns the same quotient and remainder as long division of polynomials, but with far less writing because the variables are dropped and every subtraction becomes an addition.

The one condition: the divisor must be linear with leading coefficient 1, meaning it has the form x−c. If you can read off c and list the coefficients in order, you can run the method.

Steps of Synthetic Division

Every synthetic division follows the same short loop. Read it as a way of evaluating the polynomial, not as motion to memorize.

1. Write the polynomial in standard form and insert a 0 for any missing power, so each column stands for one degree of x.

2. Find c from the divisor. Set x−c=0; for a divisor x+2 that gives c=−2.

3. List the dividend's coefficients in a row and place c to the left.

4. Bring the first coefficient straight down into the bottom row.

5. Multiply that bottom number by c and write the product under the next coefficient.

6. Add the column and write the sum in the bottom row.

7. Repeat the multiply-and-add step across every remaining column.

8. Split the bottom row: the last number is the remainder, and the numbers before it are the quotient coefficients, one degree lower than the dividend.

Examples of Synthetic Division of Polynomials

Example 1

Divide x²+x−2 by x+2.
Write the divisor as x−c, so x+2=x−(−2) and c=−2.
List the coefficients: 1, 1, −2.
Bring down the first coefficient.

1
Multiply by c=−2 and add to the next coefficient.

1×(−2)=−2 + (−2)=−1

Multiply again and add to the last coefficient.

−1×(−2)=2 ➞ 0

The bottom row is 1, −1, 0. The last number is the remainder, 0. The rest are the quotient coefficients.

Quotient=x−1, Remainder=0

Example 2

Divide 2x³+3x²−4x+12 by x−1.
Here c=1 and the coefficients are 2, 3, −4, 12.
Bring down the 2.
Multiply and add across.

2×1=2, 3+2=5
5×1=5, −4+5=1
1×1=1, 1+1=2

The bottom row is 2, 5, 1, 2.
Quotient=2x²+5x+1, Remainder=2

Example 3

Divide x³−7x+6 by x−2.
The x² term is missing.
List the visible coefficients 1, 0, −7, 6.
Multiply and add accordingly to find:

The bottom row is 1, 2, −3, 0.
Quotient=x² + 2x − 3, Remainder=0

Example 4

Test whether x−3 is a factor of x³−4x²+x+6.
Set c=3 with coefficients 1, −4, 1, 6.
Perform synthetic division to find the remainder is 0; hence x−3 is a factor.

Example 5

Divide x⁴−16 by x−2.
Insert zeros for missing terms: 1, 0, 0, 0, −16.

The bottom row is 1, 2, 4, 8, 0.
Quotient=x³ + 2x² + 4x + 8, Remainder=0

Example 6

Divide 6x² + 7x − 20 by 2x + 5.
The divisor 2x + 5 leads to c=−5/2.
Perform the steps of synthetic division.
The final quotient should be divided by 2.

Quotient=3x−4, Remainder=0

Synthetic Division vs Long Division

Why It Works

Synthetic division relates to the Remainder Theorem. When you divide p(x) by x−c, the remainder is exactly p(c).
Its efficiency is parallel to historical methods for evaluating polynomials.

Common Mistakes With Synthetic Division of a Polynomial

Mistake 1: Forgetting to insert zeros for missing terms

Mistake 2: Using the wrong sign for c

Mistake 3: Reading the last number as a quotient coefficient

Conclusion

• Synthetic division divides a polynomial by a linear factor using only coefficients and c.
• Each step follows from the last, leading to a quotient and remainder.
• Properly inserting missing zeros is crucial for accuracy.