Using Division, What Is The Quotient Of $(2x^3 - 3x - 10) \div (x - 2$\]?A. $2x^2 + 4x + 5$B. $2x^2 - 4x + 5$C. $2x^2 + 4x - 5$D. $2x^2 - 4x - 5 + \frac{1}{x-2}$

Introduction

In algebra, division of polynomials is a crucial operation that helps us simplify complex expressions and solve equations. When dividing a polynomial by another polynomial, we use long division or synthetic division to find the quotient and remainder. In this article, we will focus on using division to find the quotient of a given polynomial expression.

Understanding the Problem

The problem we are given is to find the quotient of $(2x^3 - 3x - 10) \div (x - 2)$. To solve this problem, we will use long division of polynomials. Long division is a step-by-step process that involves dividing the highest degree term of the dividend by the highest degree term of the divisor, and then multiplying the entire divisor by the result and subtracting it from the dividend.

Step 1: Set Up the Long Division

To set up the long division, we write the dividend $(2x^3 - 3x - 10)$ on top of a line, and the divisor $(x - 2)$ below it. We also write the quotient $q(x)$ on the left side of the line.

  ____________________
x - 2 | 2x^3 - 3x - 10

Step 2: Divide the Highest Degree Term

We start by dividing the highest degree term of the dividend, which is $2x^3$, by the highest degree term of the divisor, which is $x$. This gives us $2x^2$.

  2x^2
  ____________________
x - 2 | 2x^3 - 3x - 10

Step 3: Multiply the Divisor by the Result

We multiply the entire divisor $(x - 2)$ by the result $2x^2$, which gives us $2x^3 - 4x^2$.

  2x^2
  ____________________
x - 2 | 2x^3 - 3x - 10
       - (2x^3 - 4x^2)

Step 4: Subtract the Result from the Dividend

We subtract the result $2x^3 - 4x^2$ from the dividend $2x^3 - 3x - 10$, which gives us $x^2 - 3x - 10$.

  2x^2
  ____________________
x - 2 | 2x^3 - 3x - 10
       - (2x^3 - 4x^2)
       ____________________
             x^2 - 3x - 10

Step 5: Repeat the Process

We repeat the process by dividing the highest degree term of the new dividend, which is $x^2$, by the highest degree term of the divisor, which is $x$. This gives us $x$.

  2x^2 + x
  ____________________
x - 2 | 2x^3 - 3x - 10
       - (2x^3 - 4x^2)
       ____________________
             x^2 - 3x - 10
             - (x^2 - 2x)

Step 6: Subtract the Result from the Dividend

We subtract the result $x^2 - 2x$ from the dividend $x^2 - 3x - 10$, which gives us $-x - 10$.

  2x^2 + x
  ____________________
x - 2 | 2x^3 - 3x - 10
       - (2x^3 - 4x^2)
       ____________________
             x^2 - 3x - 10
             - (x^2 - 2x)
             ____________________
                   -x - 10

Step 7: Write the Quotient

We have now completed the long division process, and we can write the quotient as $2x^2 + x$.

Conclusion

In this article, we used long division to find the quotient of the polynomial expression $(2x^3 - 3x - 10) \div (x - 2)$. We set up the long division, divided the highest degree term, multiplied the divisor by the result, subtracted the result from the dividend, and repeated the process until we obtained the quotient. The final answer is $2x^2 + x$.

Answer

The correct answer is A. $2x^2 + x$.

However, we need to consider the remainder as well. The remainder is $-x - 10$, which can be written as $\frac{-x - 10}{x - 2}$. Therefore, the final answer is $2x^2 + x + \frac{-x - 10}{x - 2}$.

Alternative Answer

The alternative answer is D. $2x^2 + x - \frac{10}{x - 2}$.

Comparison of Answers

We can compare the two answers by simplifying the alternative answer. We can rewrite the alternative answer as $2x^2 + x - \frac{10}{x - 2} = 2x^2 + x - \frac{10}{x - 2} \cdot \frac{x - 2}{x - 2} = 2x^2 + x - \frac{10(x - 2)}{(x - 2)^2} = 2x^2 + x - \frac{10x - 20}{x^2 - 4x + 4} = 2x^2 + x - \frac{10x - 20}{(x - 2)^2}$.

Simplification

We can simplify the alternative answer by combining the terms. We can rewrite the alternative answer as $2x^2 + x - \frac{10x - 20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2

Introduction

In our previous article, we used long division to find the quotient of the polynomial expression $(2x^3 - 3x - 10) \div (x - 2)$. We set up the long division, divided the highest degree term, multiplied the divisor by the result, subtracted the result from the dividend, and repeated the process until we obtained the quotient. In this article, we will answer some common questions related to the topic.

Q: What is the quotient of $(2x^3 - 3x - 10) \div (x - 2)$?

A: The quotient of $(2x^3 - 3x - 10) \div (x - 2)$ is $2x^2 + x$.

Q: What is the remainder of $(2x^3 - 3x - 10) \div (x - 2)$?

A: The remainder of $(2x^3 - 3x - 10) \div (x - 2)$ is $-x - 10$.

Q: How do I simplify the remainder?

A: To simplify the remainder, you can rewrite it as $\frac{-x - 10}{x - 2}$.

Q: What is the final answer?

A: The final answer is $2x^2 + x + \frac{-x - 10}{x - 2}$.

Q: Can I use synthetic division instead of long division?

A: Yes, you can use synthetic division instead of long division. Synthetic division is a faster and more efficient method for dividing polynomials.

Q: How do I use synthetic division?

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

1. Write the coefficients of the dividend in a row.
2. Write the root of the divisor below the first coefficient.
3. Multiply the root by the first coefficient and write the result below the second coefficient.
4. Add the second coefficient and the result from step 3, and write the result below the third coefficient.
5. Repeat steps 3 and 4 until you reach the last coefficient.
6. The result is the quotient.

Q: What are the advantages of using synthetic division?

A: The advantages of using synthetic division are:

• It is faster and more efficient than long division.
• It is easier to use than long division.
• It is more accurate than long division.

Q: What are the disadvantages of using synthetic division?

A: The disadvantages of using synthetic division are:

• It is not as widely used as long division.
• It is not as well-known as long division.
• It requires more practice and experience to use effectively.

Conclusion

In this article, we answered some common questions related to the topic of using division to find the quotient of a polynomial expression. We discussed the quotient and remainder of $(2x^3 - 3x - 10) \div (x - 2)$, and how to simplify the remainder. We also discussed the advantages and disadvantages of using synthetic division instead of long division. We hope that this article has been helpful in answering your questions and providing you with a better understanding of the topic.

Final Answer

The final answer is $2x^2 + x + \frac{-x - 10}{x - 2}$.

Alternative Answer

The alternative answer is $2x^2 + x - \frac{10}{x - 2}$.

Comparison of Answers

We can compare the two answers by simplifying the alternative answer. We can rewrite the alternative answer as $2x^2 + x - \frac{10}{x - 2} = 2x^2 + x - \frac{10}{x - 2} \cdot \frac{x - 2}{x - 2} = 2x^2 + x - \frac{10(x - 2)}{(x - 2)^2} = 2x^2 + x - \frac{10x - 20}{x^2 - 4x + 4} = 2x^2 + x - \frac{10x - 20}{(x - 2)^2}$.

Simplification

We can simplify the alternative answer by combining the terms. We can rewrite the alternative answer as $2x^2 + x - \frac{10x - 20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x - \frac{10x}{(x - 2)^2} + \frac{20}{(x - 2)^2} = 2x^2 + x