What Is The Remainder In The Synthetic Division Problem Below?$1 \longdiv { 4 \, 6 \, -3 }$A. 3 B. 5 C. 7 D. 9
Introduction
Synthetic division is a method used to divide polynomials by linear factors. It is a shortcut to the long division method and is often used in algebra and calculus. In this article, we will explore the concept of synthetic division and how to find the remainder in a given problem.
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 often used in algebra and calculus. The method involves dividing the polynomial by a linear factor of the form (x - a), where a is a constant.
The Synthetic Division Process
The synthetic division process involves the following steps:
- Write down the coefficients of the polynomial in a row, with the constant term on the right.
- Bring down the first coefficient.
- Multiply the number at the bottom of the line by the number at the top of the line, and write the result below the line.
- Add the numbers in the second column, and write the result below the line.
- Repeat steps 3 and 4 until all the numbers have been processed.
- The final number in the bottom row is the remainder.
Example Problem
Let's consider the following problem:
To find the remainder, we will use the synthetic division process.
Step 1: Write Down the Coefficients
The coefficients of the polynomial are 4, 6, and -3. We will write them down in a row, with the constant term on the right.
1 | 4 6 -3
Step 2: Bring Down the First Coefficient
We will bring down the first coefficient, which is 4.
1 | 4 6 -3
| 4
Step 3: Multiply and Add
We will multiply the number at the bottom of the line (4) by the number at the top of the line (1), and write the result below the line.
1 | 4 6 -3
| 4
| 4
We will add the numbers in the second column, and write the result below the line.
1 | 4 6 -3
| 4
| 4
| 10
Step 4: Repeat the Process
We will repeat the process until all the numbers have been processed.
1 | 4 6 -3
| 4
| 4
| 10
| 14
1 | 4 6 -3
| 4
| 4
| 10
| 14
| 7
Step 5: Find the Remainder
The final number in the bottom row is the remainder. In this case, the remainder is 7.
Conclusion
In this article, we explored the concept of synthetic division and how to find the remainder in a given problem. We used the synthetic division process to find the remainder in the example problem. The final number in the bottom row is the remainder, which is 7.
Answer
The answer to the problem is:
- A. 3: Incorrect
- B. 5: Incorrect
- C. 7: Correct
- D. 9: Incorrect
Final Thoughts
Introduction
Synthetic division is a method used to divide polynomials by linear factors. It is a shortcut to the long division method and is often used in algebra and calculus. In this article, we will answer some frequently asked questions about synthetic division.
Q: What is synthetic division?
A: Synthetic division is a method of dividing polynomials by linear factors. It is a shortcut to the long division method and is often used in algebra and calculus.
Q: How does synthetic division work?
A: Synthetic division involves dividing the polynomial by a linear factor of the form (x - a), where a is a constant. The process involves bringing down the first coefficient, multiplying and adding, and repeating the process until all the numbers have been processed.
Q: What are the steps involved in synthetic division?
A: The steps involved in synthetic division are:
- Write down the coefficients of the polynomial in a row, with the constant term on the right.
- Bring down the first coefficient.
- Multiply the number at the bottom of the line by the number at the top of the line, and write the result below the line.
- Add the numbers in the second column, and write the result below the line.
- Repeat steps 3 and 4 until all the numbers have been processed.
- The final number in the bottom row is the remainder.
Q: What is the remainder in synthetic division?
A: The remainder in synthetic division is the final number in the bottom row. It is the result of the division process.
Q: How do I know if the remainder is zero?
A: If the remainder is zero, it means that the polynomial is divisible by the linear factor (x - a). In this case, the polynomial can be factored as (x - a) times the quotient.
Q: Can I use synthetic division to divide polynomials by other types of factors?
A: No, synthetic division is only used to divide polynomials by linear factors of the form (x - a). If you need to divide a polynomial by a quadratic factor or a higher-degree factor, you will need to use a different method.
Q: Is synthetic division a shortcut to the long division method?
A: Yes, synthetic division is a shortcut to the long division method. It is a faster and more efficient way to divide polynomials by linear factors.
Q: Can I use synthetic division to find the roots of a polynomial?
A: Yes, synthetic division can be used to find the roots of a polynomial. If the remainder is zero, it means that the polynomial has a root at the value of x that makes the linear factor (x - a) equal to zero.
Q: What are some common mistakes to avoid when using synthetic division?
A: Some common mistakes to avoid when using synthetic division include:
- Not bringing down the first coefficient correctly
- Not multiplying and adding correctly
- Not repeating the process until all the numbers have been processed
- Not checking the remainder to see if it is zero
Conclusion
In this article, we answered some frequently asked questions about synthetic division. We covered the basics of synthetic division, including how it works and the steps involved. We also discussed some common mistakes to avoid and how to use synthetic division to find the roots of a polynomial.
Final Thoughts
Synthetic division is a powerful tool for dividing polynomials by linear factors. It is a shortcut to the long division method and is often used in algebra and calculus. By following the steps outlined in this article, you can use synthetic division to divide polynomials and find the remainder.