Divide: $\left(6b^2 + 28b + 8\right) \div (b + 4$\]A) $6b + 4 - \frac{8}{b + 4}$ B) $6b + 2 - \frac{5}{b + 4}$ C) $6b - \frac{6}{b + 4}$ D) $6b - 1 - \frac{3}{b + 4}$ Choose The Correct Option: A B C D

Introduction

Dividing algebraic expressions can be a challenging task, especially when dealing with polynomials. However, with a clear understanding of the steps involved and a bit of practice, you can master this skill. In this article, we will focus on dividing a quadratic expression by a linear expression, using the given problem as an example.

The Problem

We are given the expression $\left(6b^2 + 28b + 8\right) \div (b + 4)$ and we need to find the quotient and remainder. To do this, we will use the long division method.

Step 1: Write the Dividend and Divisor

The dividend is the expression we want to divide, which is $6b^2 + 28b + 8$. The divisor is the expression by which we are dividing, which is $b + 4$.

Step 2: Divide the Leading Term

To start the division process, we need to divide the leading term of the dividend by the leading term of the divisor. In this case, we divide $6b^2$ by $b$, which gives us $6b$.

Step 3: Multiply and Subtract

Next, we multiply the entire divisor by the quotient we obtained in the previous step, which is $6b(b + 4) = 6b^2 + 24b$. We then subtract this product from the dividend, which gives us $6b^2 + 28b + 8 - (6b^2 + 24b) = 4b + 8$.

Step 4: Bring Down the Next Term

Since we have a remainder of $4b + 8$, we bring down the next term, which is $0$ in this case.

Step 5: Repeat the Process

We now repeat the process by dividing the leading term of the remainder by the leading term of the divisor. In this case, we divide $4b$ by $b$, which gives us $4$. We then multiply the entire divisor by this quotient, which gives us $4(b + 4) = 4b + 16$. We subtract this product from the remainder, which gives us $4b + 8 - (4b + 16) = -8$.

Step 6: Write the Quotient and Remainder

Since we have a remainder of $-8$, we can write the quotient and remainder as follows:

$$\frac{6b^2 + 28b + 8}{b + 4} = 6b + 2 - \frac{8}{b + 4}$$

Conclusion

In this article, we have shown how to divide a quadratic expression by a linear expression using the long division method. We have also provided a step-by-step guide on how to perform this division, including writing the dividend and divisor, dividing the leading term, multiplying and subtracting, bringing down the next term, and repeating the process. By following these steps, you can master the skill of dividing algebraic expressions.

Answer

The correct answer is:

• A) $6b + 4 - \frac{8}{b + 4}$ (This is not the correct answer, as we obtained $6b + 2 - \frac{8}{b + 4}$ in the previous section.)

• B) $6b + 2 - \frac{5}{b + 4}$ (This is not the correct answer, as we obtained $6b + 2 - \frac{8}{b + 4}$ in the previous section.)

• C) $6b - \frac{6}{b + 4}$ (This is not the correct answer, as we obtained $6b + 2 - \frac{8}{b + 4}$ in the previous section.)

• D) $6b - 1 - \frac{3}{b + 4}$ (This is not the correct answer, as we obtained $6b + 2 - \frac{8}{b + 4}$ in the previous section.)

The correct answer is not listed among the options. However, we can see that the correct answer is close to option B, but with a different remainder.

Discussion

This problem requires a good understanding of the long division method and the ability to perform polynomial division. It also requires the ability to write the quotient and remainder in the correct form. The problem is a good example of how to divide a quadratic expression by a linear expression, and it can be used as a reference for future problems.

Tips and Variations

• To make the problem more challenging, you can try dividing a cubic expression by a linear expression.
• To make the problem easier, you can try dividing a linear expression by a linear expression.
• To make the problem more interesting, you can try dividing a polynomial expression by a polynomial expression.

Conclusion

Introduction

Dividing algebraic expressions can be a challenging task, especially when dealing with polynomials. However, with a clear understanding of the steps involved and a bit of practice, you can master this skill. In this article, we will provide a Q&A guide on dividing algebraic expressions, including common questions and answers.

Q: What is the first step in dividing algebraic expressions?

A: The first step in dividing algebraic expressions is to write the dividend and divisor. The dividend is the expression we want to divide, and the divisor is the expression by which we are dividing.

Q: How do I divide the leading term of the dividend by the leading term of the divisor?

A: To divide the leading term of the dividend by the leading term of the divisor, we simply divide the coefficients of the leading terms. For example, if the leading term of the dividend is $6b^2$ and the leading term of the divisor is $b$, we divide $6$ by $1$ to get $6$.

Q: What is the next step in the division process?

A: The next step in the division process is to multiply the entire divisor by the quotient we obtained in the previous step and subtract the product from the dividend.

Q: How do I multiply the entire divisor by the quotient?

A: To multiply the entire divisor by the quotient, we multiply each term of the divisor by the quotient. For example, if the divisor is $b + 4$ and the quotient is $6$, we multiply $b$ by $6$ to get $6b$ and multiply $4$ by $6$ to get $24$.

Q: What is the remainder in the division process?

A: The remainder in the division process is the expression that is left over after we have divided the dividend by the divisor. It is the expression that we cannot divide further.

Q: How do I write the quotient and remainder in the correct form?

A: To write the quotient and remainder in the correct form, we write the quotient as a polynomial expression and the remainder as a fraction. For example, if the quotient is $6b + 2$ and the remainder is $-8$, we write the quotient and remainder as follows:

$$\frac{6b^2 + 28b + 8}{b + 4} = 6b + 2 - \frac{8}{b + 4}$$

Q: What are some common mistakes to avoid when dividing algebraic expressions?

A: Some common mistakes to avoid when dividing algebraic expressions include:

• Not writing the dividend and divisor correctly
• Not dividing the leading term of the dividend by the leading term of the divisor correctly
• Not multiplying the entire divisor by the quotient correctly
• Not subtracting the product from the dividend correctly
• Not writing the quotient and remainder in the correct form

Q: How can I practice dividing algebraic expressions?

A: You can practice dividing algebraic expressions by working through examples and exercises. You can also try dividing different types of expressions, such as linear expressions, quadratic expressions, and polynomial expressions.

Q: What are some real-world applications of dividing algebraic expressions?

A: Dividing algebraic expressions has many real-world applications, including:

• Science: Dividing algebraic expressions is used to solve equations and inequalities in physics, chemistry, and biology.
• Engineering: Dividing algebraic expressions is used to design and optimize systems, such as electrical circuits and mechanical systems.
• Economics: Dividing algebraic expressions is used to model and analyze economic systems, such as supply and demand curves.

Conclusion

In conclusion, dividing algebraic expressions is an important skill that requires a good understanding of the long division method and the ability to perform polynomial division. By following the steps outlined in this article and practicing dividing different types of expressions, you can master this skill and become proficient in dividing algebraic expressions.