Which Polynomial Lists The Powers In Descending Order?A. $-10 + 4x^3 - 4x^5 + 2x^4 + X^7$B. $x^7 - 4x^5 + 2x^4 + 4x^3 - 10$C. $4x^3 - 4x^5 + 2x^4 + X^7 - 10$D. $x^7 + 4x^3 + 2x^4 - 4x^5 - 10$

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Introduction

In mathematics, polynomials are algebraic expressions consisting of variables and coefficients combined using only addition, subtraction, and multiplication. When dealing with polynomials, it's essential to understand the order in which the powers of the variables are listed. In this article, we will explore which polynomial lists the powers in descending order.

Understanding Polynomials

A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. The general form of a polynomial is:

a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0

where a_n, a_(n-1), ..., a_1, a_0 are coefficients, and x is the variable.

Descending Order of Powers

The powers of the variable in a polynomial are listed in descending order when the highest power of the variable comes first, followed by the next highest power, and so on. For example, in the polynomial 3x^2 + 2x + 1, the powers of the variable x are listed in descending order: 2, 1, 0.

Analyzing the Options

Let's analyze the options given:

A. −10+4x3−4x5+2x4+x7-10 + 4x^3 - 4x^5 + 2x^4 + x^7

B. x7−4x5+2x4+4x3−10x^7 - 4x^5 + 2x^4 + 4x^3 - 10

C. 4x3−4x5+2x4+x7−104x^3 - 4x^5 + 2x^4 + x^7 - 10

D. x7+4x3+2x4−4x5−10x^7 + 4x^3 + 2x^4 - 4x^5 - 10

Option A

In option A, the powers of the variable x are listed as 7, 5, 4, 3, 0. However, the highest power of 7 comes first, followed by the next highest power of 5, and then the next highest power of 4. Therefore, the powers in option A are listed in descending order.

Option B

In option B, the powers of the variable x are listed as 7, 5, 4, 3, 0. However, the highest power of 7 comes first, followed by the next highest power of 5, and then the next highest power of 4. Therefore, the powers in option B are listed in descending order.

Option C

In option C, the powers of the variable x are listed as 3, 5, 4, 7, 0. However, the highest power of 7 does not come first, followed by the next highest power of 5, and then the next highest power of 4. Therefore, the powers in option C are not listed in descending order.

Option D

In option D, the powers of the variable x are listed as 7, 3, 4, 5, 0. However, the highest power of 7 comes first, followed by the next highest power of 5, and then the next highest power of 4. Therefore, the powers in option D are listed in descending order.

Conclusion

In conclusion, options A, B, and D list the powers of the variable x in descending order. However, option C does not list the powers in descending order. Therefore, the correct answer is options A, B, and D.

Final Answer

Introduction

In our previous article, we explored which polynomial lists the powers in descending order. In this article, we will provide a Q&A guide to help you understand the concept of polynomial powers in descending order.

Q: What is a polynomial?

A: A polynomial is an algebraic expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.

Q: What is the general form of a polynomial?

A: The general form of a polynomial is:

a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0

where a_n, a_(n-1), ..., a_1, a_0 are coefficients, and x is the variable.

Q: What is the descending order of powers?

A: The powers of the variable in a polynomial are listed in descending order when the highest power of the variable comes first, followed by the next highest power, and so on.

Q: How do I determine if the powers in a polynomial are in descending order?

A: To determine if the powers in a polynomial are in descending order, you can follow these steps:

  1. Identify the highest power of the variable in the polynomial.
  2. Check if the next highest power of the variable comes after the highest power.
  3. Continue checking the next highest powers of the variable until you reach the lowest power.

Q: What are some examples of polynomials with powers in descending order?

A: Here are some examples of polynomials with powers in descending order:

  • 3x^2 + 2x + 1
  • 2x^3 + 4x^2 + 5x + 1
  • x^4 + 2x^3 + 3x^2 + 4x + 1

Q: What are some examples of polynomials with powers not in descending order?

A: Here are some examples of polynomials with powers not in descending order:

  • 2x^2 + 3x + 4x^3
  • 4x^3 + 2x^2 + x
  • x^2 + 2x + 4x^3

Q: How do I write a polynomial with powers in descending order?

A: To write a polynomial with powers in descending order, you can follow these steps:

  1. Identify the highest power of the variable in the polynomial.
  2. Write the term with the highest power first.
  3. Continue writing the terms with the next highest powers until you reach the lowest power.

Q: What are some common mistakes to avoid when writing polynomials with powers in descending order?

A: Here are some common mistakes to avoid when writing polynomials with powers in descending order:

  • Writing the terms in the wrong order (e.g., writing the term with the lowest power first).
  • Forgetting to include a term with a specific power.
  • Including a term with a power that is not present in the polynomial.

Conclusion

In conclusion, understanding polynomial powers in descending order is an essential concept in algebra. By following the steps outlined in this article, you can determine if the powers in a polynomial are in descending order and write polynomials with powers in descending order.