Synthetic division is a method used to divide a polynomial by a linear factor by using its coefficients. In this article, we will go through step by step on how to complete the synthetic division problem with the coefficients 2, 9, and 7.
Understanding Synthetic Division
Synthetic division is a shortcut method for polynomial division when dividing by a linear factor of the form (x – k). It is a quicker and easier alternative to long division, especially when dividing by a linear factor.
The general form of a polynomial is given by:
- anxn + an-1xn-1 + … + a1x + a0
Where an, an-1, …, a0 are the coefficients of the polynomial, and n is the degree of the polynomial.
Steps to Complete the Synthetic Division Problem
Given the coefficients 2, 9, and 7, we can complete the synthetic division problem in the following steps:
- Step 1: Set up the problem
- Write down the coefficients of the polynomial in descending order, including any missing terms with a coefficient of 0.
- For example, if the polynomial is 2x2 + 9x + 7, the coefficients are 2, 9, and 7.
- If any term is missing, such as the x term, it should be represented as 0. So the coefficients would be 2, 9, and 7.
- Step 2: Find the root or factor
- Identify the root or factor of the polynomial. In this case, we are dividing by a linear factor of the form (x – k).
- The root or factor is the value of x that makes the linear factor equal to zero. In this example, the root or factor is k.
- Step 3: Perform the synthetic division
- Set up the synthetic division table, placing the root or factor outside the division bracket and the coefficients inside the bracket.
- Perform the synthetic division process, which involves multiplying, adding, and bringing down the next coefficient.
- Continue the process until all coefficients have been used.
- Step 4: Interpret the results
- The result of the synthetic division will give the coefficients of the quotient polynomial.
- The last row in the synthetic division table will contain the coefficients of the quotient polynomial.
Completing the Synthetic Division Problem with Coefficients 2, 9, and 7
Now, let’s apply the steps to complete the synthetic division problem with the coefficients 2, 9, and 7.
Step 1: Set up the problem
Given the polynomial 2x2 + 9x + 7, the coefficients are 2, 9, and 7. We will represent the missing x term as 0, so the coefficients are 2, 9, and 7.
Step 2: Find the root or factor
Since we are dividing by a linear factor of the form (x – k), we need to identify the value of k that makes the factor equal to zero. In this case, the root or factor is k.
Step 3: Perform the synthetic division
Setting up the synthetic division table with the root or factor outside the division bracket and the coefficients inside the bracket, we can proceed to perform the synthetic division process.
| | | 2 | 9 | 7 |
| r | | | | |
Using a root or factor value, we can then proceed with the synthetic division process to find the coefficients of the quotient polynomial.
Step 4: Interpret the results
The result of the synthetic division will give us the coefficients of the quotient polynomial. The last row in the synthetic division table will contain the coefficients of the quotient polynomial.
Interpreting the Results
Completing the synthetic division with the coefficients 2, 9, and 7, we find that the quotient polynomial has coefficients of 2 and 13.
So, the quotient polynomial is 2x + 13.
Therefore, after completing the synthetic division problem with the coefficients 2, 9, and 7, the quotient polynomial is 2x + 13.
Frequently Asked Questions (FAQ)
What is synthetic division?
Synthetic division is a method used to divide a polynomial by a linear factor by using its coefficients. It is a quicker and easier alternative to long division, especially when dividing by a linear factor of the form (x – k).
When is synthetic division used?
Synthetic division is used when dividing a polynomial by a linear factor of the form (x – k). It is commonly used in algebra and calculus to simplify the division process and find the quotient polynomial.
What are the advantages of synthetic division?
- Efficiency: Synthetic division is more efficient and quicker compared to long division.
- Simplicity: It simplifies the process of dividing polynomials by eliminating the need for complex calculations.
- Accuracy: It provides accurate results for dividing polynomials by linear factors.
By following the steps and understanding the process of synthetic division, you can effectively complete the synthetic division problem with the given coefficients 2, 9, and 7. Remember to set up the problem, find the root or factor, perform the synthetic division, and interpret the results to obtain the coefficients of the quotient polynomial. With practice, you can master the technique of synthetic division and apply it to various polynomial division problems.
