Select The Correct Answer.Lorie Is Using Long Division To Find The Quotient Of X 3 + 6 X 2 + 5 X^3 + 6x^2 + 5 X 3 + 6 X 2 + 5 And X 2 + X − 1 X^2 + X - 1 X 2 + X − 1 , As Shown Below:$[ \begin{aligned} x^2 + X - 1 & \quad | \quad X + 5 - \frac{4x}{x^2 + X - 1} \ & \quad

Mar 8, 2025 by ADMIN 271 views

**Select the Correct Answer: Lorie's Long Division Problem**

Understanding the Problem

When it comes to long division, it's essential to understand the process and the steps involved. In this problem, Lorie is using long division to find the quotient of two polynomials: $x^3 + 6x^2 + 5$ and $x^2 + x - 1$ . The goal is to determine the correct answer by analyzing the steps of the long division process.

The Long Division Process

Long division is a method used to divide a polynomial by another polynomial. It involves a series of steps, including dividing the leading term of the dividend by the leading term of the divisor, multiplying the entire divisor by the result, subtracting the product from the dividend, and repeating the process until the remainder is smaller than the divisor.

Analyzing Lorie's Work

Let's take a closer look at Lorie's work and see if we can identify any mistakes or areas for improvement.

{ \begin{aligned} x^2 + x - 1 & \quad | \quad x + 5 - \frac{4x}{x^2 + x - 1} \\ & \quad \end{aligned} }

In this example, Lorie has started by dividing the leading term of the dividend, $x^3$ , by the leading term of the divisor, $x^2$ . This gives a quotient of $x$ . However, Lorie has not multiplied the entire divisor by this result, which is a crucial step in the long division process.

The Correct Steps

To find the correct answer, we need to follow the correct steps of the long division process. Here's how it should be done:

Divide the leading term of the dividend, $x^3$ , by the leading term of the divisor, $x^2$ . This gives a quotient of $x$ .
Multiply the entire divisor, $x^2 + x - 1$ , by this result, $x$ . This gives a product of $x^3 + x^2 - x$ .
Subtract this product from the dividend, $x^3 + 6x^2 + 5$ . This gives a new dividend of $5x^2 + x + 5$ .
Repeat the process by dividing the leading term of the new dividend, $5x^2$ , by the leading term of the divisor, $x^2$ . This gives a quotient of $5$ .
Multiply the entire divisor, $x^2 + x - 1$ , by this result, $5$ . This gives a product of $5x^2 + 5x - 5$ .
Subtract this product from the new dividend, $5x^2 + x + 5$ . This gives a new dividend of $-4x - 0$ .
Since the new dividend is smaller than the divisor, we can stop the process and write the final answer.

The Final Answer

Based on the correct steps of the long division process, we can determine the final answer. The quotient of $x^3 + 6x^2 + 5$ and $x^2 + x - 1$ is $x + 5 - \frac{4x}{x^2 + x - 1}$ .

Conclusion

In conclusion, Lorie's long division problem requires a careful analysis of the steps involved. By following the correct steps of the long division process, we can determine the final answer and ensure that the quotient is accurate.

Key Takeaways

Long division is a method used to divide a polynomial by another polynomial.
The process involves a series of steps, including dividing the leading term of the dividend by the leading term of the divisor, multiplying the entire divisor by the result, subtracting the product from the dividend, and repeating the process until the remainder is smaller than the divisor.
It's essential to follow the correct steps of the long division process to ensure that the quotient is accurate.

Common Mistakes

Failing to multiply the entire divisor by the result.
Not subtracting the product from the dividend.
Repeating the process until the remainder is smaller than the divisor.

Tips and Tricks

Use a calculator or a computer program to check your work and ensure that the quotient is accurate.
Break down the problem into smaller steps and focus on one step at a time.
Practice, practice, practice! The more you practice, the more comfortable you'll become with the long division process.

Real-World Applications

Long division is used in a variety of real-world applications, including finance, engineering, and science.
It's used to solve problems involving polynomials, such as finding the quotient of two polynomials or solving polynomial equations.
It's also used to analyze and understand complex systems, such as electrical circuits or mechanical systems.

Conclusion

In conclusion, long division is a powerful tool used to divide a polynomial by another polynomial. By following the correct steps of the long division process, we can determine the final answer and ensure that the quotient is accurate. With practice and patience, anyone can master the long division process and apply it to a variety of real-world problems.

Understanding the Basics

Long division of polynomials is a fundamental concept in algebra that involves dividing a polynomial by another polynomial. It's a crucial skill to master, as it's used in a variety of real-world applications, including finance, engineering, and science.

Q: What is long division of polynomials?

A: Long division of polynomials is a method used to divide a polynomial by another polynomial. It involves a series of steps, including dividing the leading term of the dividend by the leading term of the divisor, multiplying the entire divisor by the result, subtracting the product from the dividend, and repeating the process until the remainder is smaller than the divisor.

Q: Why is long division of polynomials important?

A: Long division of polynomials is important because it's used to solve problems involving polynomials, such as finding the quotient of two polynomials or solving polynomial equations. It's also used to analyze and understand complex systems, such as electrical circuits or mechanical systems.

Q: What are the steps involved in long division of polynomials?

A: The steps involved in long division of polynomials are:

Divide the leading term of the dividend by the leading term of the divisor.
Multiply the entire divisor by the result.
Subtract the product from the dividend.
Repeat the process until the remainder is smaller than the divisor.

Q: What are some common mistakes to avoid when performing long division of polynomials?

A: Some common mistakes to avoid when performing long division of polynomials include:

Failing to multiply the entire divisor by the result.
Not subtracting the product from the dividend.
Repeating the process until the remainder is smaller than the divisor.

Q: How can I practice long division of polynomials?

A: You can practice long division of polynomials by using a calculator or a computer program to check your work and ensure that the quotient is accurate. You can also practice by breaking down the problem into smaller steps and focusing on one step at a time.

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

A: Some real-world applications of long division of polynomials include:

Finance: Long division of polynomials is used to calculate interest rates and investment returns.
Engineering: Long division of polynomials is used to design and analyze complex systems, such as electrical circuits or mechanical systems.
Science: Long division of polynomials is used to solve problems involving polynomials, such as finding the quotient of two polynomials or solving polynomial equations.

Q: Can I use a calculator or computer program to perform long division of polynomials?

A: Yes, you can use a calculator or computer program to perform long division of polynomials. This can be helpful if you're struggling with the process or if you need to check your work.

Q: How can I improve my skills in long division of polynomials?

A: You can improve your skills in long division of polynomials by practicing regularly and breaking down the problem into smaller steps. You can also use online resources or work with a tutor to get additional help.

Conclusion

In conclusion, long division of polynomials is a fundamental concept in algebra that involves dividing a polynomial by another polynomial. By understanding the basics and practicing regularly, you can master the long division process and apply it to a variety of real-world problems.

Key Takeaways

Long division of polynomials is a method used to divide a polynomial by another polynomial.
The process involves a series of steps, including dividing the leading term of the dividend by the leading term of the divisor, multiplying the entire divisor by the result, subtracting the product from the dividend, and repeating the process until the remainder is smaller than the divisor.
It's essential to follow the correct steps of the long division process to ensure that the quotient is accurate.

Common Mistakes

Failing to multiply the entire divisor by the result.
Not subtracting the product from the dividend.
Repeating the process until the remainder is smaller than the divisor.

Tips and Tricks

Use a calculator or a computer program to check your work and ensure that the quotient is accurate.
Break down the problem into smaller steps and focus on one step at a time.
Practice, practice, practice! The more you practice, the more comfortable you'll become with the long division process.

Real-World Applications

Long division of polynomials is used in a variety of real-world applications, including finance, engineering, and science.
It's used to solve problems involving polynomials, such as finding the quotient of two polynomials or solving polynomial equations.
It's also used to analyze and understand complex systems, such as electrical circuits or mechanical systems.