Example of Descending Order

Oct 13, 2025 by leolavanderia.com.br 117 views

Introduction

In algebra, polynomials are mathematical expressions consisting of variables and coefficients combined using addition, subtraction, and multiplication. When dealing with polynomials, it's essential to understand the concept of order, particularly descending order. In this article, we will explore the concept of descending order in polynomials and examine the given options to determine which one shows the polynomial written in descending order.

What is Descending Order in Polynomials?

Descending order in polynomials refers to the arrangement of terms from highest degree to lowest degree. In other words, the term with the highest power of the variable is written first, followed by the term with the next highest power, and so on. This arrangement is crucial in algebra, as it allows for easier manipulation and simplification of polynomials.

Example of Descending Order

To illustrate the concept of descending order, let's consider a simple polynomial: $2x^2 + 3x + 1$ . In this polynomial, the term with the highest power of the variable is $2x^2$ , followed by the term $3x$ , and finally the constant term $1$ . Therefore, the polynomial is written in descending order.

The Given Polynomial

The given polynomial is: $5x^3 - x + 9x^7 + 4 + 3x^{11}$ .

Analyzing the Options

Let's examine the given options to determine which one shows the polynomial written in descending order.

Option A

$3x^{11} + 9x^7 - x + 4 + 5x^3$

In this option, the term with the highest power of the variable is $3x^{11}$ , followed by the term $9x^7$ , and then the term $5x^3$ . However, the term $-x$ is not in its correct position, as it should be placed before the constant term $4$ . Therefore, this option is not correct.

Option B

$3x^{11} + 9x^7 + 5x^3 - x + 4$

Option C

$9x^7 + 5x^3 + 4 + 3x^{11} - x$

In this option, the term with the highest power of the variable is $9x^7$ , followed by the term $5x^3$ , and then the term $3x^{11}$ . However, the term $-x$ is not in its correct position, as it should be placed before the constant term $4$ . Therefore, this option is not correct.

Option D

$3x^{11} + 9x^7 + 5x^3 - x + 4$

In this option, the term with the highest power of the variable is $3x^{11}$ , followed by the term $9x^7$ , and then the term $5x^3$ . The term $-x$ is in its correct position, as it is placed before the constant term $4$ . Therefore, this option is correct.

Conclusion

In conclusion, the correct option that shows the polynomial written in descending order is Option D: $3x^{11} + 9x^7 + 5x^3 - x + 4$ . This option correctly arranges the terms from highest degree to lowest degree, making it the correct answer.

Final Answer

$The final answer is: $\boxed{D}$<br/> **Polynomial Order: Understanding Descending Order in Algebra**$

Q&A: Frequently Asked Questions about Polynomial Order

Q: What is the importance of descending order in polynomials? A: Descending order in polynomials is crucial in algebra as it allows for easier manipulation and simplification of polynomials. It also helps in identifying the degree of the polynomial, which is essential in various mathematical operations.

Q: How do I determine the descending order of a polynomial? A: To determine the descending order of a polynomial, you need to arrange the terms from highest degree to lowest degree. The term with the highest power of the variable is written first, followed by the term with the next highest power, and so on.

Q: What is the difference between ascending and descending order in polynomials? A: Ascending order in polynomials refers to the arrangement of terms from lowest degree to highest degree, whereas descending order refers to the arrangement of terms from highest degree to lowest degree.

Q: Can you provide an example of a polynomial in descending order? A: Yes, consider the polynomial: $2x^2 + 3x + 1$ . In this polynomial, the term with the highest power of the variable is $2x^2$ , followed by the term $3x$ , and finally the constant term $1$ . Therefore, the polynomial is written in descending order.

Q: How do I identify the degree of a polynomial? A: The degree of a polynomial is determined by the highest power of the variable in the polynomial. For example, in the polynomial $2x^2 + 3x + 1$ , the degree is 2, as the highest power of the variable is 2.

Q: Can you provide an example of a polynomial in ascending order? A: Yes, consider the polynomial: $1 + 3x + 2x^2$ . In this polynomial, the term with the lowest power of the variable is $1$ , followed by the term $3x$ , and finally the term $2x^2$ . Therefore, the polynomial is written in ascending order.

Q: What is the significance of the constant term in a polynomial? A: The constant term in a polynomial is the term that does not contain the variable. It is essential in various mathematical operations, such as addition and subtraction of polynomials.

Q: Can you provide an example of a polynomial with a constant term? A: Yes, consider the polynomial: $2x^2 + 3x + 1$ . In this polynomial, the constant term is $1$ , which is the term that does not contain the variable.

Q: How do I simplify a polynomial? A: To simplify a polynomial, you need to combine like terms, which are terms that have the same variable and exponent. For example, in the polynomial $2x^2 + 3x + 1$ , you can combine the like terms $2x^2$ and $3x$ to get $2x^2 + 3x$ .

Q: Can you provide an example of a polynomial that is already in descending order? A: Yes, consider the polynomial: $3x^{11} + 9x^7 + 5x^3 - x + 4$ . In this polynomial, the term with the highest power of the variable is $3x^{11}$ , followed by the term $9x^7$ , and then the term $5x^3$ . The term $-x$ is in its correct position, as it is placed before the constant term $4$ . Therefore, the polynomial is already in descending order.

Conclusion

In conclusion, understanding descending order in polynomials is crucial in algebra. By following the steps outlined in this article, you can determine the descending order of a polynomial and simplify it. Remember, the term with the highest power of the variable is written first, followed by the term with the next highest power, and so on.

Final Answer

The final answer is: $\boxed{D}$