Consider The Quotient: \left(x^3-8x+6\right) \div \left(x^2-2x+1\right ].Use Long Division To Rewrite The Quotient In An Equivalent Form As Q ( X ) + R ( X ) B ( X ) Q(x) + \frac{r(x)}{b(x)} Q ( X ) + B ( X ) R ( X ) ​ , Where Q ( X Q(x Q ( X ] Is The Quotient, R ( X R(x R ( X ] Is The

Introduction

When it comes to dividing polynomials, long division is a powerful tool that can help us simplify complex expressions and rewrite them in an equivalent form. In this article, we will explore the process of long division and how it can be used to rewrite the quotient in an equivalent form as $q(x) + \frac{r(x)}{b(x)}$. We will use the given polynomial $\left(x^3-8x+6\right) \div \left(x^2-2x+1\right)$ as an example to illustrate the process.

Understanding the Problem

Before we begin the long division process, it's essential to understand the problem and what we are trying to achieve. We are given a polynomial $\left(x^3-8x+6\right)$ that we want to divide by another polynomial $\left(x^2-2x+1\right)$. Our goal is to rewrite the quotient in an equivalent form as $q(x) + \frac{r(x)}{b(x)}$, where $q(x)$ is the quotient, $r(x)$ is the remainder, and $b(x)$ is the divisor.

The Long Division Process

To perform long division, we need to follow a series of steps that involve dividing the polynomial and finding the remainder. Here's a step-by-step guide to the long division process:

Step 1: Divide the Leading Term

The first step in long division is to divide the leading term of the dividend by the leading term of the divisor. In this case, we need to divide $x^3$ by $x^2$. This will give us the first term of the quotient, which is $x$.

Step 2: Multiply and Subtract

Once we have the first term of the quotient, we need to multiply the entire divisor by this term and subtract the result from the dividend. In this case, we need to multiply $\left(x^2-2x+1\right)$ by $x$ and subtract the result from $\left(x^3-8x+6\right)$. This will give us a new polynomial that we can use to continue the division process.

Step 3: Bring Down the Next Term

After subtracting the result from the previous step, we need to bring down the next term of the dividend. In this case, we need to bring down the term $-8x$.

Step 4: Repeat the Process

We need to repeat the process of dividing the leading term of the new polynomial by the leading term of the divisor, multiplying and subtracting, and bringing down the next term. We will continue this process until we have divided all the terms of the dividend.

Applying the Long Division Process

Now that we have a step-by-step guide to the long division process, let's apply it to the given polynomial $\left(x^3-8x+6\right) \div \left(x^2-2x+1\right)$. We will follow the steps outlined above to perform the long division.

Step 1: Divide the Leading Term

The first step is to divide the leading term of the dividend by the leading term of the divisor. In this case, we need to divide $x^3$ by $x^2$. This will give us the first term of the quotient, which is $x$.

Step 2: Multiply and Subtract

Once we have the first term of the quotient, we need to multiply the entire divisor by this term and subtract the result from the dividend. In this case, we need to multiply $\left(x^2-2x+1\right)$ by $x$ and subtract the result from $\left(x^3-8x+6\right)$. This will give us a new polynomial that we can use to continue the division process.

Step 3: Bring Down the Next Term

After subtracting the result from the previous step, we need to bring down the next term of the dividend. In this case, we need to bring down the term $-8x$.

Step 4: Repeat the Process

We need to repeat the process of dividing the leading term of the new polynomial by the leading term of the divisor, multiplying and subtracting, and bringing down the next term. We will continue this process until we have divided all the terms of the dividend.

Finding the Quotient and Remainder

After performing the long division process, we will have a quotient and a remainder. The quotient will be a polynomial that represents the result of dividing the dividend by the divisor, and the remainder will be a polynomial that represents the amount left over after the division.

Rewriting the Quotient in an Equivalent Form

Once we have the quotient and remainder, we can rewrite the quotient in an equivalent form as $q(x) + \frac{r(x)}{b(x)}$. This will give us a new expression that represents the result of the division in a different form.

Conclusion

In this article, we have explored the process of long division and how it can be used to rewrite the quotient in an equivalent form as $q(x) + \frac{r(x)}{b(x)}$. We have applied the long division process to the given polynomial $\left(x^3-8x+6\right) \div \left(x^2-2x+1\right)$ and found the quotient and remainder. We have also rewritten the quotient in an equivalent form as $q(x) + \frac{r(x)}{b(x)}$. This has given us a new expression that represents the result of the division in a different form.

Final Answer

The final answer is $\boxed{x - 2 + \frac{4x - 5}{x^2 - 2x + 1}}$.

References

• [1] "Polynomial Long Division" by Math Open Reference
• [2] "Long Division of Polynomials" by Purplemath
• [3] "Dividing Polynomials" by Khan Academy
  

Introduction

Dividing polynomials can be a challenging task, but with the right tools and techniques, it can be made easier. In this article, we will answer some of the most frequently asked questions about dividing polynomials, including the long division process, finding the quotient and remainder, and rewriting the quotient in an equivalent form.

Q: What is the long division process for dividing polynomials?

A: The long division process for dividing polynomials involves dividing the leading term of the dividend by the leading term of the divisor, multiplying and subtracting, and bringing down the next term. This process is repeated until all the terms of the dividend have been divided.

Q: How do I find the quotient and remainder when dividing polynomials?

A: To find the quotient and remainder when dividing polynomials, you need to perform the long division process. The quotient will be a polynomial that represents the result of dividing the dividend by the divisor, and the remainder will be a polynomial that represents the amount left over after the division.

Q: What is the difference between the quotient and the remainder?

A: The quotient is a polynomial that represents the result of dividing the dividend by the divisor, while the remainder is a polynomial that represents the amount left over after the division. The quotient and remainder are related by the equation: dividend = quotient × divisor + remainder.

Q: How do I rewrite the quotient in an equivalent form as q(x) + r(x)/b(x)?

A: To rewrite the quotient in an equivalent form as q(x) + r(x)/b(x), you need to take the quotient and remainder from the long division process and rewrite them in the desired form. This involves dividing the remainder by the divisor and adding the result to the quotient.

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

A: Some common mistakes to avoid when dividing polynomials include:

• Not performing the long division process correctly
• Not finding the correct quotient and remainder
• Not rewriting the quotient in the correct form
• Not checking the work for errors

Q: How do I check my work when dividing polynomials?

A: To check your work when dividing polynomials, you need to multiply the quotient by the divisor and add the remainder. If the result is equal to the dividend, then your work is correct. If not, then you need to recheck your work and make any necessary corrections.

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

A: Dividing polynomials has many real-world applications, including:

• Algebra: Dividing polynomials is a fundamental concept in algebra, and it is used to solve equations and inequalities.
• Calculus: Dividing polynomials is used in calculus to find the derivative and integral of a function.
• Engineering: Dividing polynomials is used in engineering to design and analyze systems.
• Computer Science: Dividing polynomials is used in computer science to develop algorithms and data structures.

Q: How do I practice dividing polynomials?

A: To practice dividing polynomials, you can try the following:

• Use online resources, such as Khan Academy or Mathway, to practice dividing polynomials.
• Work with a partner or tutor to practice dividing polynomials.
• Use real-world examples to practice dividing polynomials.
• Create your own problems to practice dividing polynomials.

Conclusion

Dividing polynomials can be a challenging task, but with the right tools and techniques, it can be made easier. By following the steps outlined in this article, you can learn how to divide polynomials and apply this knowledge to real-world problems. Remember to practice regularly and to check your work for errors.

Final Answer

The final answer is $\boxed{q(x) + r(x)/b(x)}$.

References

• [1] "Polynomial Long Division" by Math Open Reference
• [2] "Long Division of Polynomials" by Purplemath
• [3] "Dividing Polynomials" by Khan Academy