Activity
In this activity we’re going to use synthetic division to divide a polynomial by a binomial.
Part 1: Use Long Division
You are going to solve a division problem using long division.
1.
Compare your answer:
Part 2: Use Synthetic Division
You are now going to solve the same division problem using synthetic division.
2.
Compare your answer:
Step 1 - Write the dividend with decreasing powers of . Make sure there is no degree missing from the highest degree down to a degree of .
Step 2 - Write the coefficients of the terms as the first row of the synthetic division.
Step 3 - Write the divisor as and place in the synthetic division in the divisor box.
Step 4 - Bring down the first coefficient to the third row.
Step 5 - Multiply that coefficient by the divisor and place the result in the second row under the second coefficient.
Step 6 - Add the second column, putting the result in the third row.
Step 7 - Multiply that result by the divisor and place the result in the second row under the third coefficient.
Step 8 - Add the final column, putting the result in the third row.
Step 9 - The division is complete. The numbers in the third row give us the result. The in the third row are the coefficients of the quotient. The quotient is . The in the box in the third row is the remainder.
Work with a partner to find each quotient. Discuss any differences you have in determining the correct solution.
3.
Compare your answer:
; remainder of 1
4.
Compare your answer:
; remainder of 4
5.
Compare your answer:
; remainder of 0
6.
Compare your answer:
; remainder of 0
Self Check
Additional Resources
Dividing Polynomials Using Synthetic Division
Mathematicians like to find patterns to make their work easier. Since long division can be tedious, let’s look back at the long division we did in a previous example and look for some patterns. We will use this as a basis for what is called synthetic division. The same problem in the synthetic division format is shown next.
Synthetic division basically just removes unnecessary repeated variables and numbers. Here, all the and are removed as well as the and because they are opposite the term above.
The first row of the synthetic division is the coefficients of the dividend. The is the opposite of the in the divisor.
The second row of the synthetic division is the numbers shown in red in the division problem.
The third row of the synthetic division is the numbers shown in blue in the division problem.
Notice the quotient and remainder are shown in the third row.
Synthetic division only works when the divisor is of the form .
The following example will explain the process.
Example 1
Use synthetic division to find the quotient and remainder when is divided by .
Step 1 - Write the dividend with decreasing powers of .
Step 2 - Write the coefficients of the terms as the first row of the synthetic division.
Step 3 - Write the divisor as and place in the synthetic division in the divisor box.
Step 4 - Bring down the first coefficient to the third row.
Step 5 - Multiply that coefficient by the divisor and place the result in the second row under the second coefficient.
Step 6 - Add the second column, putting the result in the third row.
Step 7 - Multiply that result by the divisor and place the result in the second row under the third coefficient.
Step 8 - Add the third column, putting the result in the third row.
Step 9 - Multiply that result by the divisor and place the result in the third row under the third coefficient.
Step 10 - Add the final column, putting the result in the third row.
Step 11 - The quotient is , and the remainder is .
The division is complete. The numbers in the third row give us the result. The are the coefficients of the quotient. The quotient is . The in the box in the third row is the remainder.
Compare your answer:
Example 2
Use synthetic division to find the quotient and remainder when is divided by .
The polynomial has its terms in order with descending degree, but we notice there is no term. We will add a as a placeholder for the term. In form, the divisor is .
We divided a 4th degree polynomial by a 1st degree polynomial, so the quotient will be a 3rd degree polynomial. Reading from the third row, the quotient has the coefficients , which is . The remainder is .
Try it
Try It: Dividing Polynomials Using Synthetic Division
1. Use synthetic division to find the quotient and remainder when is divided by .
2. Use synthetic division to find the quotient and remainder when is divided by .
Here is how to find these quotients using synthetic division:
1. ; remainder of
2. ; remainder of