In order to divide polynomials using synthetic division, you must be dividing by a linear expression and the leading coefficient (first number) must be a 1. For example, you can use synthetic division to divide by x + 3 or x – 6, but you cannot use synthetic division to divide by x^{2} + 2 or 3x^{2} – x + 7. If the leading coefficient is not a 1, then you must divide by the leading coefficient to turn the leading coefficient into a 1. For example, 3x – 1 would become and 2x + 7 would become. If synthetic division will not work, then you must use long division.
Here are the steps required for Synthetic Division of a Polynomial:
Step 1 : | To set up the problem, first, set the denominator equal to zero to find the number to put in the division box. Next, make sure the numerator is written in descending order and if any terms are missing you must use a zero to fill in the missing term, finally list only the coefficient in the division problem. |
Step 2 : | Once the problem is set up correctly, bring the leading coefficient (first number) straight down. |
Step 3 : | Multiply the number in the division box with the number you brought down and put the result in the next column. |
Step 4 : | Add the two numbers together and write the result in the bottom of the row. |
Step 5 : | Repeat steps 3 and 4 until you reach the end of the problem. |
Step 6 : | Write the final answer. The final answer is made up of the numbers in the bottom row with the last number being the remainder and the remainder must be written as a fraction. The variables or x’s start off one power less than the original denominator and go down one with each term. |
Example 1 - Divide:
Step 1: To set up the problem, first, set the denominator equal to zero to find the number to put in the division box. Next, make sure the numerator is written in descending order and if any terms are missing you must use a zero to fill in the missing term, finally list only the coefficient in the division problem. | |
Step 2: Once the problem is set up correctly, bring the leading coefficient (first number) straight down. | |
Step 3: Multiply the number in the division box with the number you brought down and put the result in the next column. | |
Step 4: Add the two numbers together and write the result in the bottom of the row. | |
Step 5: Multiply the number in the division box with the number you brought down and put the result in the next column. | |
Step 6: Add the two numbers together and write the result in the bottom of the row. | |
Step 7: Multiply the number in the division box with the number you brought down and put the result in the next column. | |
Step 8: Add the two numbers together and write the result in the bottom of the row. | |
Step 9: Write the final answer. The final answer is made up of the numbers in the bottom row with the last number being the remainder and the remainder must be written as a fraction. The variables or x’s start off one power less than the original denominator and go down one with each term. |
Example 2 - Divide:
Step 1: To set up the problem, first, set the denominator equal to zero to find the number to put in the division box. Next, make sure the numerator is written in descending order and if any terms are missing you must use a zero to fill in the missing term, finally list only the coefficient in the division problem. | |
Step 2: Once the problem is set up correctly, bring the leading coefficient (first number) straight down. | |
Step 3: Multiply the number in the division box with the number you brought down and put the result in the next column. | |
Step 4: Add the two numbers together and write the result in the bottom of the row. | |
Step 5: Multiply the number in the division box with the number you brought down and put the result in the next column. | |
Step 6: Add the two numbers together and write the result in the bottom of the row. | |
Step 7: Multiply the number in the division box with the number you brought down and put the result in the next column. | |
Step 8: Add the two numbers together and write the result in the bottom of the row. | |
Step 9: Write the final answer. The final answer is made up of the numbers in the bottom row with the last number being the remainder and the remainder must be written as a fraction. The variables or x’s start off one power less than the original denominator and go down one with each term. |
Click Here for Practice Problems
Example 3 – Divide:
Step 1: To set up the problem, first, set the denominator equal to zero to find the number to put in the division box. Next, make sure the numerator is written in descending order and if any terms are missing you must use a zero to fill in the missing term, finally list only the coefficient in the division problem. | |
Step 2: Once the problem is set up correctly, bring the leading coefficient (first number) straight down. | |
Step 3: Multiply the number in the division box with the number you brought down and put the result in the next column. | |
Step 4: Add the two numbers together and write the result in the bottom of the row. | |
Step 5: Repeat steps 3 and 4 until you reach the end of the problem. | |
Step 6 : Write the final answer. The final answer is made up of the numbers in the bottom row with the last number being the remainder and the remainder must be written as a fraction. The variables or x’s start off one power less than the original denominator and go down one with each term. |
Click Here for Practice Problems
Example 4 – Divide:
Step 1: To set up the problem, first, set the denominator equal to zero to find the number to put in the division box. Next, make sure the numerator is written in descending order and if any terms are missing you must use a zero to fill in the missing term, finally list only the coefficient in the division problem. | |
Step 2: Once the problem is set up correctly, bring the leading coefficient (first number) straight down. | |
Step 3: Multiply the number in the division box with the number you brought down and put the result in the next column. | |
Step 4: Add the two numbers together and write the result in the bottom of the row. | |
Step 5: Repeat steps 3 and 4 until you reach the end of the problem. | |
Step 6 : Write the final answer. The final answer is made up of the numbers in the bottom row with the last number being the remainder and the remainder must be written as a fraction. The variables or x’s start off one power less than the original denominator and go down one with each term. |
Click Here for Practice Problems
Example 5 – Divide:
Step 1: To set up the problem, first, set the denominator equal to zero to find the number to put in the division box. Next, make sure the numerator is written in descending order and if any terms are missing you must use a zero to fill in the missing term, finally list only the coefficient in the division problem. | |
Step 2: Once the problem is set up correctly, bring the leading coefficient (first number) straight down. | |
Step 3: Multiply the number in the division box with the number you brought down and put the result in the next column. | |
Step 4: Add the two numbers together and write the result in the bottom of the row. | |
Step 5: Repeat steps 3 and 4 until you reach the end of the problem. | |
Step 6 : Write the final answer. The final answer is made up of the numbers in the bottom row with the last number being the remainder and the remainder must be written as a fraction. The variables or x’s start off one power less than the original denominator and go down one with each term. |