Divide The Following Expression:$ \frac{4x^3 - 2x^2 - X}{x - 1} }$Find The Quotient And Remainder ${ x^2 + \square X + \square + \frac{\square {x - 1} }$

Introduction

In algebra, polynomial division is a crucial concept that helps us simplify complex expressions and find the quotient and remainder. In this article, we will focus on dividing the given expression $\frac{4x^3 - 2x^2 - x}{x - 1}$ and finding the quotient and remainder in the form $x^2 + \square x + \square + \frac{\square}{x - 1}$. We will break down the division process into manageable steps and provide a clear explanation of each step.

Understanding Polynomial Division

Before we dive into the division process, let's briefly review the concept of polynomial division. Polynomial division is the process of dividing a polynomial by another polynomial, resulting in a quotient and a remainder. The dividend is the polynomial being divided, and the divisor is the polynomial by which we are dividing. The quotient is the result of the division, and the remainder is the amount left over after the division.

Setting Up the Division

To divide the given expression, we need to set up the division in the correct format. We will divide the polynomial $4x^3 - 2x^2 - x$ by the polynomial $x - 1$. We will use long division to perform the division.

Performing the Division

To perform the division, we will follow these steps:

1. Divide the leading term of the dividend by the leading term of the divisor: We will divide the leading term of the dividend, $4x^3$, by the leading term of the divisor, $x$. This gives us $4x^2$.
2. Multiply the divisor by the result: We will multiply the divisor, $x - 1$, by the result, $4x^2$. This gives us $4x^3 - 4x^2$.
3. Subtract the result from the dividend: We will subtract the result, $4x^3 - 4x^2$, from the dividend, $4x^3 - 2x^2 - x$. This gives us $2x^2 - x$.
4. Repeat the process: We will repeat the process with the new dividend, $2x^2 - x$, and the divisor, $x - 1$.

Finding the Quotient and Remainder

After performing the division, we will have the quotient and remainder in the form $x^2 + \square x + \square + \frac{\square}{x - 1}$. We will identify the values of the coefficients and the remainder.

Calculating the Quotient and Remainder

Let's perform the division and find the quotient and remainder.

Step 1: Divide the leading term of the dividend by the leading term of the divisor

We will divide the leading term of the dividend, $4x^3$, by the leading term of the divisor, $x$. This gives us $4x^2$.

Step 2: Multiply the divisor by the result

We will multiply the divisor, $x - 1$, by the result, $4x^2$. This gives us $4x^3 - 4x^2$.

Step 3: Subtract the result from the dividend

We will subtract the result, $4x^3 - 4x^2$, from the dividend, $4x^3 - 2x^2 - x$. This gives us $2x^2 - x$.

Step 4: Repeat the process

We will repeat the process with the new dividend, $2x^2 - x$, and the divisor, $x - 1$.

Step 5: Divide the leading term of the new dividend by the leading term of the divisor

We will divide the leading term of the new dividend, $2x^2$, by the leading term of the divisor, $x$. This gives us $2x$.

Step 6: Multiply the divisor by the result

We will multiply the divisor, $x - 1$, by the result, $2x$. This gives us $2x^2 - 2x$.

Step 7: Subtract the result from the new dividend

We will subtract the result, $2x^2 - 2x$, from the new dividend, $2x^2 - x$. This gives us $x$.

Step 8: Repeat the process

We will repeat the process with the new dividend, $x$, and the divisor, $x - 1$.

Step 9: Divide the leading term of the new dividend by the leading term of the divisor

We will divide the leading term of the new dividend, $x$, by the leading term of the divisor, $x$. This gives us $1$.

Step 10: Multiply the divisor by the result

We will multiply the divisor, $x - 1$, by the result, $1$. This gives us $x - 1$.

Step 11: Subtract the result from the new dividend

We will subtract the result, $x - 1$, from the new dividend, $x$. This gives us $2$.

Conclusion

After performing the division, we have found the quotient and remainder in the form $x^2 + \square x + \square + \frac{\square}{x - 1}$. The quotient is $x^2 + 2x + 1$, and the remainder is $\frac{2}{x - 1}$.

Final Answer

The final answer is $\boxed{x^2 + 2x + 1 + \frac{2}{x - 1}}$.

Introduction

In the previous article, we divided the given expression $\frac{4x^3 - 2x^2 - x}{x - 1}$ and found the quotient and remainder in the form $x^2 + \square x + \square + \frac{\square}{x - 1}$. In this article, we will answer some frequently asked questions (FAQs) about dividing the given expression.

Q: What is the purpose of dividing the given expression?

A: The purpose of dividing the given expression is to simplify the complex expression and find the quotient and remainder.

Q: What is the quotient and remainder in the form $x^2 + \square x + \square + \frac{\square}{x - 1}$?

A: The quotient is $x^2 + 2x + 1$, and the remainder is $\frac{2}{x - 1}$.

Q: How do I perform the division?

A: To perform the division, you need to follow the steps outlined in the previous article. You will need to divide the leading term of the dividend by the leading term of the divisor, multiply the divisor by the result, subtract the result from the dividend, and repeat the process until you have the quotient and remainder.

Q: What is the significance of the remainder?

A: The remainder is the amount left over after the division. In this case, the remainder is $\frac{2}{x - 1}$, which means that the original expression cannot be simplified further.

Q: Can I use other methods to divide the given expression?

A: Yes, you can use other methods to divide the given expression, such as synthetic division or polynomial long division. However, the steps outlined in the previous article are a more straightforward and efficient way to perform the division.

Q: How do I check my work?

A: To check your work, you can multiply the quotient by the divisor and add the remainder to the product. If the result is equal to the original expression, then your work is correct.

Q: What are some common mistakes to avoid when dividing the given expression?

A: Some common mistakes to avoid when dividing the given expression include:

• Not following the correct order of operations
• Not multiplying the divisor by the result
• Not subtracting the result from the dividend
• Not repeating the process until you have the quotient and remainder

Q: Can I use the result of the division in other mathematical operations?

A: Yes, you can use the result of the division in other mathematical operations, such as addition, subtraction, multiplication, and division.

Q: How do I apply the result of the division in real-world problems?

A: The result of the division can be applied in real-world problems, such as:

• Finding the area of a circle
• Finding the volume of a sphere
• Finding the surface area of a cylinder
• Finding the volume of a cone

Conclusion

In this article, we have answered some frequently asked questions (FAQs) about dividing the given expression. We have provided a clear and concise explanation of the division process and the significance of the quotient and remainder. We have also provided some common mistakes to avoid and some real-world applications of the result of the division.

Final Answer

The final answer is $\boxed{x^2 + 2x + 1 + \frac{2}{x - 1}}$.