Divide The Expression:${ \left(12s^3 + 3s^2 - 9s\right) \div (3s + 3) }$

Introduction

When it comes to simplifying algebraic expressions, division can be a challenging operation. However, with a clear understanding of the rules and a step-by-step approach, you can easily divide complex expressions like the one given in this problem. In this article, we will guide you through the process of dividing the expression $\left(12s^3 + 3s^2 - 9s\right) \div (3s + 3)$ and provide you with a clear understanding of the underlying concepts.

Understanding the Expression

Before we dive into the division process, let's take a closer look at the given expression. The expression consists of two parts: the dividend $\left(12s^3 + 3s^2 - 9s\right)$ and the divisor $(3s + 3)$. Our goal is to simplify the expression by dividing the dividend by the divisor.

The Division Process

To divide the expression, we will use the long division method. This method involves dividing the highest degree term of the dividend by the highest degree term of the divisor, and then multiplying the result by the divisor and subtracting the product from the dividend. We will repeat this process until we have simplified the expression.

Step 1: Divide the Highest Degree Term

The highest degree term of the dividend is $12s^3$, and the highest degree term of the divisor is $3s$. To divide these two terms, we will divide $12s^3$ by $3s$, which gives us $4s^2$.

Step 2: Multiply and Subtract

Now that we have divided the highest degree term, we will multiply the result by the divisor and subtract the product from the dividend. The product of $4s^2$ and $(3s + 3)$ is $12s^3 + 12s^2$. Subtracting this product from the dividend gives us $-9s^2 - 9s$.

Step 3: Repeat the Process

We will repeat the process of dividing the highest degree term of the dividend by the highest degree term of the divisor, multiplying the result by the divisor, and subtracting the product from the dividend. The highest degree term of the dividend is now $-9s^2$, and the highest degree term of the divisor is still $3s$. Dividing $-9s^2$ by $3s$ gives us $-3s$.

Step 4: Multiply and Subtract Again

Multiplying $-3s$ by the divisor $(3s + 3)$ gives us $-9s^2 - 9s$. Subtracting this product from the dividend $-9s^2 - 9s$ gives us $0$.

The Final Result

After repeating the process of dividing the highest degree term of the dividend by the highest degree term of the divisor, multiplying the result by the divisor, and subtracting the product from the dividend, we have simplified the expression to $4s^2 - 3s$.

Conclusion

Dividing the expression $\left(12s^3 + 3s^2 - 9s\right) \div (3s + 3)$ using the long division method involves dividing the highest degree term of the dividend by the highest degree term of the divisor, multiplying the result by the divisor, and subtracting the product from the dividend. By repeating this process, we can simplify the expression and arrive at the final result of $4s^2 - 3s$.

Frequently Asked Questions

• What is the long division method? The long division method is a step-by-step process for dividing one polynomial by another. It involves dividing the highest degree term of the dividend by the highest degree term of the divisor, multiplying the result by the divisor, and subtracting the product from the dividend.
• How do I divide a polynomial by a binomial? To divide a polynomial by a binomial, you will use the long division method. This involves dividing the highest degree term of the dividend by the highest degree term of the divisor, multiplying the result by the divisor, and subtracting the product from the dividend.
• What is the final result of dividing the expression $\left(12s^3 + 3s^2 - 9s\right) \div (3s + 3)$? The final result of dividing the expression $\left(12s^3 + 3s^2 - 9s\right) \div (3s + 3)$ is $4s^2 - 3s$.

Tips and Tricks

• Make sure to divide the highest degree term of the dividend by the highest degree term of the divisor. This will ensure that you are dividing the largest possible terms and simplifying the expression as much as possible.
• Multiply the result by the divisor and subtract the product from the dividend. This will help you to eliminate the terms that are being divided and simplify the expression.
• Repeat the process until you have simplified the expression. This will ensure that you have divided the entire expression and arrived at the final result.

Related Topics

• Polynomial Division: Polynomial division is the process of dividing one polynomial by another. It involves dividing the highest degree term of the dividend by the highest degree term of the divisor, multiplying the result by the divisor, and subtracting the product from the dividend.
• Binomial Division: Binomial division is a type of polynomial division where the divisor is a binomial. It involves dividing the highest degree term of the dividend by the highest degree term of the divisor, multiplying the result by the divisor, and subtracting the product from the dividend.
• Algebraic Expressions: Algebraic expressions are mathematical expressions that involve variables and constants. They can be simplified using various techniques, including polynomial division.

Conclusion

Dividing the expression $\left(12s^3 + 3s^2 - 9s\right) \div (3s + 3)$ using the long division method involves dividing the highest degree term of the dividend by the highest degree term of the divisor, multiplying the result by the divisor, and subtracting the product from the dividend. By repeating this process, we can simplify the expression and arrive at the final result of $4s^2 - 3s$.

Introduction

Dividing algebraic expressions can be a challenging task, but with a clear understanding of the rules and a step-by-step approach, you can easily simplify complex expressions. In this article, we will answer some of the most frequently asked questions about dividing algebraic expressions.

Q&A

Q: What is the long division method?

A: The long division method is a step-by-step process for dividing one polynomial by another. It involves dividing the highest degree term of the dividend by the highest degree term of the divisor, multiplying the result by the divisor, and subtracting the product from the dividend.

Q: How do I divide a polynomial by a binomial?

A: To divide a polynomial by a binomial, you will use the long division method. This involves dividing the highest degree term of the dividend by the highest degree term of the divisor, multiplying the result by the divisor, and subtracting the product from the dividend.

Q: What is the final result of dividing the expression $\left(12s^3 + 3s^2 - 9s\right) \div (3s + 3)$?

A: The final result of dividing the expression $\left(12s^3 + 3s^2 - 9s\right) \div (3s + 3)$ is $4s^2 - 3s$.

Q: How do I simplify a complex expression using polynomial division?

A: To simplify a complex expression using polynomial division, you will use the long division method. This involves dividing the highest degree term of the dividend by the highest degree term of the divisor, multiplying the result by the divisor, and subtracting the product from the dividend.

Q: What is the difference between polynomial division and binomial division?

A: Polynomial division is the process of dividing one polynomial by another, while binomial division is a type of polynomial division where the divisor is a binomial.

Q: How do I handle negative numbers when dividing algebraic expressions?

A: When dividing algebraic expressions, you will handle negative numbers by following the rules of arithmetic. If the dividend is negative, you will change the sign of the result. If the divisor is negative, you will change the sign of the result.

Q: Can I use the long division method to divide a polynomial by a trinomial?

A: Yes, you can use the long division method to divide a polynomial by a trinomial. However, you will need to follow the rules of polynomial division and handle the trinomial as a single divisor.

Q: How do I check my work when dividing algebraic expressions?

A: To check your work when dividing algebraic expressions, you will multiply the result by the divisor and subtract the product from the dividend. If the result is zero, you have correctly simplified the expression.

Tips and Tricks

• Make sure to divide the highest degree term of the dividend by the highest degree term of the divisor. This will ensure that you are dividing the largest possible terms and simplifying the expression as much as possible.
• Multiply the result by the divisor and subtract the product from the dividend. This will help you to eliminate the terms that are being divided and simplify the expression.
• Repeat the process until you have simplified the expression. This will ensure that you have divided the entire expression and arrived at the final result.

Related Topics

• Polynomial Division: Polynomial division is the process of dividing one polynomial by another. It involves dividing the highest degree term of the dividend by the highest degree term of the divisor, multiplying the result by the divisor, and subtracting the product from the dividend.
• Binomial Division: Binomial division is a type of polynomial division where the divisor is a binomial. It involves dividing the highest degree term of the dividend by the highest degree term of the divisor, multiplying the result by the divisor, and subtracting the product from the dividend.
• Algebraic Expressions: Algebraic expressions are mathematical expressions that involve variables and constants. They can be simplified using various techniques, including polynomial division.

Conclusion

Dividing algebraic expressions can be a challenging task, but with a clear understanding of the rules and a step-by-step approach, you can easily simplify complex expressions. By following the long division method and handling negative numbers and trinomials correctly, you can arrive at the final result of $4s^2 - 3s$.