Rewriting the Final Remainder

Oct 13, 2025 by windailys.com 56 views

Introduction

Polynomial division is a fundamental concept in algebra that involves dividing one polynomial by another. It is a crucial operation in mathematics, with numerous applications in various fields, including engineering, physics, and computer science. In this article, we will explore the division of polynomials, focusing on the work shown for the division problem. We will also delve into the discussion category of mathematics, providing a comprehensive understanding of the concept.

The Division Problem

The given division problem is:

\frac{2x^4 - 4x^3 - 11x^2 + 3x - 6}{x + 2}

Our goal is to divide the polynomial $2x^4 - 4x^3 - 11x^2 + 3x - 6$ by $x + 2$ . To do this, we will use the long division method, which involves dividing the leading term of the dividend by the leading term of the divisor.

The Long Division Method

The long division method involves the following steps:

Divide the leading term of the dividend by the leading term of the divisor.
Multiply the entire divisor by the result from step 1.
Subtract the product from step 2 from the dividend.
Bring down the next term of the dividend.
Repeat steps 1-4 until the degree of the remainder is less than the degree of the divisor.

The Work Shown

Let's apply the long division method to the given division problem:

\begin{array}{r|rrrrr} -2 & 2 & -4 & -11 & 3 & -6 \\ & & -4 & 16 & -10 & 14 \end{array}

In the first step, we divide the leading term of the dividend, $2x^4$ , by the leading term of the divisor, $x$ . This gives us $2x^3$ . We multiply the entire divisor, $x + 2$ , by $2x^3$ , which gives us $2x^4 + 4x^3$ . We subtract this product from the dividend, $2x^4 - 4x^3 - 11x^2 + 3x - 6$ , resulting in $-8x^3 - 11x^2 + 3x - 6$ .

The Remainder

The remainder is the result of the subtraction in the previous step. In this case, the remainder is:

-8x^3 - 11x^2 + 3x - 6

However, we are asked to find the remainder in the form of a box. To do this, we need to rewrite the remainder as a polynomial of degree less than the degree of the divisor. In this case, the degree of the divisor is 1, so we need to rewrite the remainder as a polynomial of degree 0.

Rewriting the Remainder

To rewrite the remainder as a polynomial of degree 0, we can divide the leading term of the remainder, $-8x^3$ , by the leading term of the divisor, $x$ . This gives us $-8x^2$ . We multiply the entire divisor, $x + 2$ , by $-8x^2$ , which gives us $-8x^3 - 16x^2$ . We subtract this product from the remainder, $-8x^3 - 11x^2 + 3x - 6$ , resulting in $5x^2 + 3x - 6$ .

The Final Remainder

The final remainder is the result of the subtraction in the previous step. In this case, the final remainder is:

5x^2 + 3x - 6

However, we are asked to find the remainder in the form of a box. To do this, we can rewrite the final remainder as a polynomial of degree 0.

Rewriting the Final Remainder

To rewrite the final remainder as a polynomial of degree 0, we can divide the leading term of the final remainder, $5x^2$ , by the leading term of the divisor, $x$ . This gives us $5x$ . We multiply the entire divisor, $x + 2$ , by $5x$ , which gives us $5x^2 + 10x$ . We subtract this product from the final remainder, $5x^2 + 3x - 6$ , resulting in $-7x - 6$ .

The Final Answer

The final remainder is the result of the subtraction in the previous step. In this case, the final remainder is:

-7x - 6

However, we are asked to find the remainder in the form of a box. To do this, we can rewrite the final remainder as a polynomial of degree 0.

Rewriting the Final Remainder

To rewrite the final remainder as a polynomial of degree 0, we can divide the leading term of the final remainder, $-7x$ , by the leading term of the divisor, $x$ . This gives us $-7$ . We multiply the entire divisor, $x + 2$ , by $-7$ , which gives us $-7x - 14$ . We subtract this product from the final remainder, $-7x - 6$ , resulting in $\boxed{8}$ .

Conclusion

In this article, we explored the division of polynomials, focusing on the work shown for the division problem. We applied the long division method to the given division problem and obtained the remainder in the form of a box. We also rewrote the remainder as a polynomial of degree 0, resulting in the final answer of $\boxed{8}$ .

Discussion Category: Mathematics

The division of polynomials is a fundamental concept in mathematics, with numerous applications in various fields. It is a crucial operation in algebra, with many real-world applications. In this article, we provided a comprehensive understanding of the concept, including the long division method and the work shown for the division problem.

Key Takeaways

The division of polynomials is a fundamental concept in mathematics.
The long division method involves dividing the leading term of the dividend by the leading term of the divisor.
The remainder is the result of the subtraction in the long division method.
The remainder can be rewritten as a polynomial of degree 0.
The final remainder is the result of the subtraction in the long division method.

References

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

Introduction

In our previous article, we explored the division of polynomials, focusing on the work shown for the division problem. We applied the long division method to the given division problem and obtained the remainder in the form of a box. In this article, we will provide a Q&A section to further clarify any doubts or questions that readers may have.

Q1: What is the long division method?

A1: The long division method is a step-by-step process for dividing one polynomial by another. It involves dividing the leading term of the dividend by the leading term of the divisor, then multiplying the entire divisor by the result and subtracting the product from the dividend.

Q2: How do I apply the long division method?

A2: To apply the long division method, follow these steps:

Divide the leading term of the dividend by the leading term of the divisor.
Multiply the entire divisor by the result from step 1.
Subtract the product from step 2 from the dividend.
Bring down the next term of the dividend.
Repeat steps 1-4 until the degree of the remainder is less than the degree of the divisor.

Q3: What is the remainder in the long division method?

A3: The remainder is the result of the subtraction in the long division method. It is the amount left over after dividing the dividend by the divisor.

Q4: How do I rewrite the remainder as a polynomial of degree 0?

A4: To rewrite the remainder as a polynomial of degree 0, divide the leading term of the remainder by the leading term of the divisor. Multiply the entire divisor by the result and subtract the product from the remainder.

Q5: What is the final remainder in the long division method?

A5: The final remainder is the result of the subtraction in the long division method. It is the amount left over after dividing the dividend by the divisor.

Q6: How do I apply the long division method to a division problem?

A6: To apply the long division method to a division problem, follow these steps:

Write the dividend and divisor in the long division format.
Divide the leading term of the dividend by the leading term of the divisor.
Multiply the entire divisor by the result from step 2.
Subtract the product from step 3 from the dividend.
Bring down the next term of the dividend.
Repeat steps 2-5 until the degree of the remainder is less than the degree of the divisor.

Q7: What are some common mistakes to avoid in the long division method?

A7: Some common mistakes to avoid in the long division method include:

Not following the correct order of operations.
Not multiplying the entire divisor by the result.
Not subtracting the product from the dividend.
Not bringing down the next term of the dividend.

Q8: How do I check my work in the long division method?

A8: To check your work in the long division method, follow these steps:

Multiply the entire divisor by the result from the long division method.
Subtract the product from the dividend.
If the result is zero, then the long division method is correct.

Conclusion

In this article, we provided a Q&A section to further clarify any doubts or questions that readers may have about the division of polynomials. We covered topics such as the long division method, the remainder, and how to apply the long division method to a division problem. We also discussed common mistakes to avoid and how to check your work in the long division method.

Discussion Category: Mathematics

Key Takeaways

The long division method involves dividing the leading term of the dividend by the leading term of the divisor.
The remainder is the result of the subtraction in the long division method.
The remainder can be rewritten as a polynomial of degree 0.
The final remainder is the result of the subtraction in the long division method.
Common mistakes to avoid in the long division method include not following the correct order of operations, not multiplying the entire divisor by the result, not subtracting the product from the dividend, and not bringing down the next term of the dividend.

References

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