Select The Correct Answer.Using Synthetic Division, What Is The Factored Form Of This Polynomial?$3x^3 - 16x^2 + 12x + 16$A. $(3x - 2)(x + 4)(x + 2)$B. $ ( 3 X − 2 ) ( X − 4 ) ( X − 2 ) (3x - 2)(x - 4)(x - 2) ( 3 X − 2 ) ( X − 4 ) ( X − 2 ) [/tex]C. $(3x + 2)(x + 4)(x +
Introduction to Synthetic Division
Synthetic division is a powerful technique used to divide polynomials and find their roots. It is a shortcut method that simplifies the process of polynomial division, making it easier to factor polynomials and find their roots. In this article, we will explore how to use synthetic division to factor a given polynomial.
The Given Polynomial
The given polynomial is:
Step 1: Set Up the Synthetic Division Table
To use synthetic division, we need to set up a table with the coefficients of the polynomial. The coefficients are the numbers that multiply the powers of x. In this case, the coefficients are 3, -16, 12, and 16.
| 3 | -16 | 12 | 16 |
|---|
Step 2: Choose a Root
To use synthetic division, we need to choose a root to divide the polynomial by. The root should be a number that we think might be a factor of the polynomial. In this case, we will choose x = 2 as our root.
Step 3: Perform the Synthetic Division
Now, we will perform the synthetic division by dividing the polynomial by x - 2.
| 3 | -16 | 12 | 16 | |
|---|---|---|---|---|
| 2 | 6 | -12 | 24 | 32 |
Step 4: Write the Result
After performing the synthetic division, we get the following result:
Step 5: Factor the Quadratic
Now, we need to factor the quadratic expression 3x^2 - 4x - 8. We can use factoring methods such as grouping or the quadratic formula to factor the quadratic.
Step 6: Factor the Quadratic Using Grouping
Let's use the grouping method to factor the quadratic.
Step 7: Write the Final Factored Form
Now, we can write the final factored form of the polynomial.
Conclusion
In this article, we used synthetic division to factor a given polynomial. We set up the synthetic division table, chose a root, performed the synthetic division, and wrote the result. We then factored the quadratic expression using the grouping method and wrote the final factored form of the polynomial.
Discussion
The final factored form of the polynomial is:
This is the correct answer.
Comparison with Other Options
Let's compare our answer with the other options.
Option A: $(3x - 2)(x + 4)(x + 2)$
This is not the correct answer because the signs of the factors are not correct.
Option B: $(3x - 2)(x - 4)(x - 2)$
This is not the correct answer because the signs of the factors are not correct.
Final Answer
The final answer is:
(x - 2)(3x + 2)(x - 4)$<br/>
# Synthetic Division: Q&A
Synthetic division is a powerful technique used to divide polynomials and find their roots. In this article, we will answer some common questions about synthetic division and provide examples to help illustrate the concepts. A: Synthetic division is a shortcut method of polynomial division that simplifies the process of dividing polynomials and finding their roots. A: To set up the synthetic division table, you need to write the coefficients of the polynomial in a row, with the highest degree term first. Then, you need to write the root of the polynomial in a column to the left of the coefficients. A: The synthetic division table is used to perform the division of the polynomial by the root. The table helps you to keep track of the coefficients of the quotient and the remainder. A: You can choose any root that you think might be a factor of the polynomial. However, if you choose a root that is not a factor of the polynomial, the synthetic division will not work. A: Synthetic division is a shortcut method of polynomial division that simplifies the process of dividing polynomials and finding their roots. Long division is a more traditional method of polynomial division that involves dividing the polynomial by the root using a series of steps. A: Yes, you can use synthetic division to divide polynomials with complex roots. However, you need to use the complex conjugate of the root as the divisor. A: You can check the synthetic division by multiplying the quotient by the divisor and adding the remainder. If the result is equal to the original polynomial, then the synthetic division is correct. A: Yes, you can use synthetic division to factor polynomials with multiple roots. However, you need to use the synthetic division multiple times, each time using a different root. A: Some common mistakes to avoid when using synthetic division include: In this article, we have answered some common questions about synthetic division and provided examples to help illustrate the concepts. We have also discussed some common mistakes to avoid when using synthetic division. Synthetic division is a powerful technique used to divide polynomials and find their roots. It is a shortcut method that simplifies the process of polynomial division and makes it easier to factor polynomials and find their roots. The final answer is: Synthetic division is a powerful technique used to divide polynomials and find their roots. It is a shortcut method that simplifies the process of polynomial division and makes it easier to factor polynomials and find their roots.Introduction
Q: What is synthetic division?
Q: How do I set up the synthetic division table?
Q: What is the purpose of the synthetic division table?
Q: How do I choose a root for synthetic division?
Q: What is the difference between synthetic division and long division?
Q: Can I use synthetic division to divide polynomials with complex roots?
Q: How do I know if the synthetic division is correct?
Q: Can I use synthetic division to factor polynomials with multiple roots?
Q: What are some common mistakes to avoid when using synthetic division?
Conclusion
Discussion
Final Answer
Additional Resources