Synthetic division is a method of polynomial division that is commonly used in algebra. It is often used to divide a polynomial by a linear factor of the form (x-c), where c is a constant. In this tutorial, we will walk through how to complete the synthetic division problem below 2 7 5 step by step.
Understanding Synthetic Division
Synthetic division is a shorthand method for dividing polynomials, particularly when dividing by linear factors. It is a quick and efficient method that requires less writing and fewer calculations compared to long division. Synthetic division can be used to find the quotient and remainder when dividing a polynomial by a linear factor. The procedure involves using the coefficients of the polynomial and the divisor to carry out a sequence of simple arithmetic operations.
Step-by-Step Guide to Complete the Synthetic Division Problem Below 2 7 5
Now, let’s go through the step-by-step process of completing the synthetic division problem with the coefficients 2, 7, and 5.
- Write down the coefficients: Start by writing down the coefficients of the polynomial in the synthetic division format. In this case, the coefficients are 2, 7, and 5.
- Identify the divisor: Identify the linear factor that you will be dividing the polynomial by. The linear factor should be of the form (x-c), where c is a constant.
- Set up the synthetic division table: Write down the divisor to the left and the coefficients to the right in a row. For our example, the linear factor is 2, so we set up the table as follows:
| 2 | | | 2 | 7 | 5 |
Now, we can proceed with the division process.
- Bring down the first coefficient: Write down the first coefficient (2) below the division bar.
- Multiply and add: Multiply the divisor by the number below the division bar, and write the result above the bar. Then, add the two numbers and write the sum below the bar.
- Repeat the process: Continue the process with the new result until all coefficients have been processed.
Completing the Synthetic Division
Let’s walk through the steps of completing the synthetic division for the coefficients 2, 7, and 5 using the divisor 2.
| 2 | | | 2 | 7 | 5 | |
| 2 | 18 | 50 | |||
| 2 | 9 | 25 |
After completing the synthetic division, the results are 2, 9, and 25. These numbers correspond to the coefficients of the quotient polynomial and the remainder. The quotient polynomial is obtained by taking the coefficients obtained after the division, while the remainder is the last number obtained after the division.
Verifying the Results
To verify the results, we can reconstruct the original polynomial using the quotient and remainder obtained from the synthetic division. In this case, our quotient polynomial is 2x^2 + 9x + 25, and the remainder is 0. We can express the original polynomial as:
2x^2 + 7x + 5 = (x – 2)(2x^2 + 9x + 25)
By performing the multiplication, we should obtain the original polynomial. This step helps to confirm that the synthetic division was performed correctly and that the quotient and remainder are accurate.
FAQs
Q: What is the purpose of synthetic division?
A: Synthetic division is used to divide polynomials, particularly when dividing by linear factors of the form (x-c). It helps to find the quotient and remainder efficiently, especially when dealing with linear factors.
Q: When should synthetic division be used?
A: Synthetic division should be used when dividing polynomials by linear factors, as it offers a quicker and more straightforward method compared to long division. It is particularly helpful when dealing with linear factors of the form (x-c).
Q: What are the steps to complete synthetic division?
A: The steps to complete synthetic division include writing down the coefficients, identifying the divisor, setting up the synthetic division table, bringing down the first coefficient, multiplying and adding, and repeating the process until all coefficients have been processed.
