Use Synthetic Division To Find The Quotient And The Remainder Of The Following Expression: $ \frac{2b^4 - 10b^3 + 30b + 9}{b - 4} }$Choose The Correct Answer A. { Q(b) = 2b^4 - 2b^3 - 8b^2 - 2b + 1 $ B . \[ B. \[ B . \[ Q(b) = 2b^3 +

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 a linear factor of the form (x - c) or (b - c). In this article, we will use synthetic division to find the quotient and the remainder of the expression $\frac{2b^4 - 10b^3 + 30b + 9}{b - 4}$.

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 a linear factor of the form (x - c) or (b - c). The method involves using a single row of numbers to perform the division, rather than the multiple rows required by the long division method.

How to Perform Synthetic Division

To perform synthetic division, we need to follow these steps:

1. Write down the coefficients of the polynomial in a row, with the constant term on the right.
2. Write down the value of the linear factor (in this case, b - 4) on the left.
3. Bring down the first coefficient (in this case, 2).
4. Multiply the value of the linear factor by the first coefficient and write the result below the second coefficient.
5. Add the second coefficient and the result from step 4.
6. Repeat steps 4 and 5 for each coefficient.
7. The final result is the quotient and the remainder.

Example: Using Synthetic Division to Find the Quotient and Remainder

Let's use synthetic division to find the quotient and the remainder of the expression $\frac{2b^4 - 10b^3 + 30b + 9}{b - 4}$.

	2	-10	0	30	9
4	2	-42	0	120	36

Step 1: Bring down the first coefficient

The first coefficient is 2. We bring it down to the next row.

	2	-42	0	120	36

Step 2: Multiply the value of the linear factor by the first coefficient

The value of the linear factor is 4. We multiply it by the first coefficient (2) and write the result below the second coefficient.

	2	-42	0	120	36
4	8	-42	0	120	36

Step 3: Add the second coefficient and the result from step 2

The second coefficient is -10. We add it to the result from step 2 (-42).

	2	-42	0	120	36
4	8	-84	0	120	36

Step 4: Repeat steps 2 and 3 for each coefficient

We repeat steps 2 and 3 for each coefficient.

	2	-42	0	120	36
4	8	-84	0	120	36
	2	-42	0	120	36
4	8	-84	0	120	36
	2	-42	0	120	36
4	8	-84	0	120	36
	2	-42	0	120	36
4	8	-84	0	120	36

Step 5: The final result is the quotient and the remainder

The final result is the quotient and the remainder.

Quotient: $2b^3 - 2b^2 - 8b - 2$ Remainder: 1

Conclusion

In this article, we used synthetic division to find the quotient and the remainder of the expression $\frac{2b^4 - 10b^3 + 30b + 9}{b - 4}$. We followed the steps of synthetic division and obtained the quotient and the remainder. The quotient is $2b^3 - 2b^2 - 8b - 2$ and the remainder is 1.

Answer

The correct answer is:

A. $Q(b) = 2b^3 - 2b^2 - 8b - 2$

Discussion

Synthetic division is a powerful tool for polynomial division. It is a shortcut to the long division method and is particularly useful when dividing polynomials by a linear factor of the form (x - c) or (b - c). In this article, we used synthetic division to find the quotient and the remainder of the expression $\frac{2b^4 - 10b^3 + 30b + 9}{b - 4}$. We followed the steps of synthetic division and obtained the quotient and the remainder. The quotient is $2b^3 - 2b^2 - 8b - 2$ and the remainder is 1.

References

• [1] "Synthetic Division" by Math Open Reference. Retrieved 2023-02-20.
• [2] "Polynomial Division" by Khan Academy. Retrieved 2023-02-20.

Keywords

• Synthetic division
• Polynomial division
• Quotient
• Remainder
• Linear factor
• Polynomial
• Division
• Math
• Algebra
  Synthetic Division: A Q&A Guide =====================================

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 a linear factor of the form (x - c) or (b - c). In this article, we will answer some 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 a linear factor of the form (x - c) or (b - c).

Q: How do I perform synthetic division?

A: To perform synthetic division, you need to follow these steps:

1. Write down the coefficients of the polynomial in a row, with the constant term on the right.
2. Write down the value of the linear factor (in this case, b - 4) on the left.
3. Bring down the first coefficient (in this case, 2).
4. Multiply the value of the linear factor by the first coefficient and write the result below the second coefficient.
5. Add the second coefficient and the result from step 4.
6. Repeat steps 4 and 5 for each coefficient.
7. The final result is the quotient and the remainder.

Q: What is the quotient and remainder in synthetic division?

A: The quotient is the result of dividing the polynomial by the linear factor, and the remainder is the amount left over after the division.

Q: How do I find the quotient and remainder in synthetic division?

A: To find the quotient and remainder in synthetic division, you need to follow the steps outlined above. The quotient is the result of dividing the polynomial by the linear factor, and the remainder is the amount left over after the division.

Q: What are some common mistakes to avoid in synthetic division?

A: Some common mistakes to avoid in synthetic division include:

• Not writing down the coefficients of the polynomial in the correct order.
• Not writing down the value of the linear factor correctly.
• Not bringing down the first coefficient correctly.
• Not multiplying the value of the linear factor by the first coefficient correctly.
• Not adding the second coefficient and the result from step 4 correctly.

Q: How do I check my work in synthetic division?

A: To check your work in synthetic division, you need to follow these steps:

1. Multiply the value of the linear factor by the first coefficient and write the result below the second coefficient.
2. Add the second coefficient and the result from step 1.
3. Repeat steps 1 and 2 for each coefficient.
4. The final result is the quotient and the remainder.

Q: What are some real-world applications of synthetic division?

A: Synthetic division has many real-world applications, including:

• Dividing polynomials in algebra.
• Finding the roots of a polynomial.
• Solving systems of equations.
• Finding the maximum or minimum of a function.

Q: How do I use synthetic division to solve systems of equations?

A: To use synthetic division to solve systems of equations, you need to follow these steps:

1. Write down the coefficients of the polynomial in a row, with the constant term on the right.
2. Write down the value of the linear factor (in this case, b - 4) on the left.
3. Bring down the first coefficient (in this case, 2).
4. Multiply the value of the linear factor by the first coefficient and write the result below the second coefficient.
5. Add the second coefficient and the result from step 4.
6. Repeat steps 4 and 5 for each coefficient.
7. The final result is the quotient and the remainder.

Conclusion

In this article, we answered some frequently asked questions about synthetic division. We covered topics such as how to perform synthetic division, what the quotient and remainder are, and how to check your work. We also discussed some real-world applications of synthetic division and how to use it to solve systems of equations.

References

• [1] "Synthetic Division" by Math Open Reference. Retrieved 2023-02-20.
• [2] "Polynomial Division" by Khan Academy. Retrieved 2023-02-20.

Keywords

• Synthetic division
• Polynomial division
• Quotient
• Remainder
• Linear factor
• Polynomial
• Division
• Math
• Algebra
• Systems of equations
• Real-world applications