College

Which polynomial lists the powers in descending order?

A. [tex]10x^2 + 8x^3 + x^8 - 2 + 3x^6[/tex]

B. [tex]3x^6 + 10x^2 + x^8 + 8x^3 - 2[/tex]

C. [tex]x^8 + 3x^6 + 8x^3 + 10x^2 - 2[/tex]

D. [tex]x^8 + 10x^2 + 8x^3 + 3x^6 - 2[/tex]

Answer :

To arrange a polynomial in descending order, we need to list the terms starting from the highest power of the variable to the lowest. Let's break down the process for each option:

1. Option A:
- [tex]\( 10x^2 \)[/tex]
- [tex]\( 8x^3 \)[/tex]
- [tex]\( x^8 \)[/tex]
- Constant [tex]\( -2 \)[/tex]
- [tex]\( 3x^6 \)[/tex]

Here, the highest power term is [tex]\( x^8 \)[/tex], and it is not the first term, so this isn't in descending order.

2. Option B:
- [tex]\( 3x^6 \)[/tex]
- [tex]\( 10x^2 \)[/tex]
- [tex]\( x^8 \)[/tex]
- [tex]\( 8x^3 \)[/tex]
- Constant [tex]\( -2 \)[/tex]

Again, the term [tex]\( x^8 \)[/tex] should come first as it has the highest power, but it doesn't, so this is not in descending order.

3. Option C:
- [tex]\( x^8 \)[/tex]
- [tex]\( 3x^6 \)[/tex]
- [tex]\( 8x^3 \)[/tex]
- [tex]\( 10x^2 \)[/tex]
- Constant [tex]\( -2 \)[/tex]

This option begins with the highest power, [tex]\( x^8 \)[/tex], followed by lower powers in sequence down to the constant term. This matches the criteria for descending order.

4. Option D:
- [tex]\( x^8 \)[/tex]
- [tex]\( 10x^2 \)[/tex]
- [tex]\( 8x^3 \)[/tex]
- [tex]\( 3x^6 \)[/tex]
- Constant [tex]\( -2 \)[/tex]

Although it starts with [tex]\( x^8 \)[/tex], the order of powers after that is not continuously descending, so this isn't in descending order.

Based on the above analysis, Option C lists the powers of the polynomial in descending order.

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