Pregunta 3Divide Using Synthetic Division:$\left(16x^4 + 4x^3 + 2x^2 - 21x + 7\right) \div (4x - 1$\]A. $4x^3 + 2x^2 + X - 5 + \frac{2}{4x - 1}$B. $4x^3 + 2x^2 + X - 5$C. $x^3 + X^2 + X$Pregunta 4Divide Using
Introduction
Synthetic division is a method used to divide polynomials by linear factors. It is a shortcut to the long division method and is particularly useful when dividing polynomials by factors of the form (x - a) or (ax + b). In this article, we will use synthetic division to divide the polynomial by the linear factor .
What is Synthetic Division?
Synthetic division is a method of dividing polynomials by linear factors. It is a shortcut to the long division method and is particularly useful when dividing polynomials by factors of the form (x - a) or (ax + b). The method involves setting up a table with the coefficients of the polynomial and the root of the linear factor. The table is then used to calculate the coefficients of the quotient and the remainder.
How to Perform Synthetic Division
To perform synthetic division, we need to follow these steps:
- Set up the table with the coefficients of the polynomial and the root of the linear factor.
- Bring down the first coefficient of the polynomial.
- Multiply the root of the linear factor by the first coefficient and add the result to the second coefficient.
- Multiply the root of the linear factor by the result from step 3 and add the result to the third coefficient.
- Continue this process until we have calculated all the coefficients of the quotient and the remainder.
Example: Divide Using Synthetic Division
Let's use synthetic division to divide the polynomial by the linear factor .
Step 1: Set Up the Table
| 1 | 4 | 2 | -21 | 7 | |
|---|---|---|---|---|---|
| 4x - 1 | 4 | 16 | 8 | -85 | 35 |
Step 2: Bring Down the First Coefficient
The first coefficient of the polynomial is 16. We bring it down to the next row.
Step 3: Multiply the Root of the Linear Factor by the First Coefficient and Add the Result to the Second Coefficient
The root of the linear factor is 1/4. We multiply it by the first coefficient (16) and add the result to the second coefficient (4).
(16) × (1/4) = 4 4 + 4 = 8
Step 4: Multiply the Root of the Linear Factor by the Result from Step 3 and Add the Result to the Third Coefficient
We multiply the root of the linear factor (1/4) by the result from step 3 (8) and add the result to the third coefficient (2).
(8) × (1/4) = 2 2 + 2 = 4
Step 5: Continue the Process
We continue the process until we have calculated all the coefficients of the quotient and the remainder.
| 1 | 4 | 2 | -21 | 7 | |
|---|---|---|---|---|---|
| 4x - 1 | 4 | 16 | 8 | -85 | 35 |
The final result is:
Quotient: Remainder:
Conclusion
In this article, we used synthetic division to divide the polynomial by the linear factor . We set up a table with the coefficients of the polynomial and the root of the linear factor and used it to calculate the coefficients of the quotient and the remainder. The final result is .
Answer
The correct answer is:
Frequently Asked Questions About Synthetic Division
Q: What is synthetic division?
A: Synthetic division is a method used to divide polynomials by linear factors. It is a shortcut to the long division method and is particularly useful when dividing polynomials by factors of the form (x - a) or (ax + b).
Q: When should I use synthetic division?
A: You should use synthetic division when dividing polynomials by linear factors. It is particularly useful when the linear factor is of the form (x - a) or (ax + b).
Q: How do I perform synthetic division?
A: To perform synthetic division, you need to follow these steps:
- Set up the table with the coefficients of the polynomial and the root of the linear factor.
- Bring down the first coefficient of the polynomial.
- Multiply the root of the linear factor by the first coefficient and add the result to the second coefficient.
- Multiply the root of the linear factor by the result from step 3 and add the result to the third coefficient.
- Continue this process until you have calculated all the coefficients of the quotient and the remainder.
Q: What is the difference between synthetic division and long division?
A: Synthetic division is a shortcut to the long division method. It is faster and more efficient than long division, but it requires the linear factor to be of the form (x - a) or (ax + b).
Q: Can I use synthetic division to divide polynomials by factors of the form (x^2 + ax + b)?
A: No, you cannot use synthetic division to divide polynomials by factors of the form (x^2 + ax + b). Synthetic division is only used to divide polynomials by linear factors of the form (x - a) or (ax + b).
Q: How do I know if I have used synthetic division correctly?
A: To check if you have used synthetic division correctly, you can multiply the quotient by the linear factor and add the remainder. If the result is the original polynomial, then you have used synthetic division correctly.
Q: What are some common mistakes to avoid when using synthetic division?
A: Some common mistakes to avoid when using synthetic division include:
- Not setting up the table correctly
- Not bringing down the first coefficient correctly
- Not multiplying the root of the linear factor by the correct coefficient
- Not adding the result to the correct coefficient
Q: Can I use synthetic division to divide polynomials with complex coefficients?
A: Yes, you can use synthetic division to divide polynomials with complex coefficients. However, you need to be careful when multiplying complex numbers and adding the result to the correct coefficient.
Q: How do I use synthetic division to divide polynomials with negative coefficients?
A: You can use synthetic division to divide polynomials with negative coefficients by following the same steps as before. However, you need to be careful when multiplying negative numbers and adding the result to the correct coefficient.
Conclusion
In this article, we have answered some frequently asked questions about synthetic division. We have covered topics such as when to use synthetic division, how to perform synthetic division, and common mistakes to avoid. We hope that this article has been helpful in understanding synthetic division and how to use it to divide polynomials.