To determine the standard form of the given polynomial, we need to ensure that the terms are ordered by the power of [tex]\( x \)[/tex] in descending order. Here are the given options:
1. [tex]\(\frac{x^3}{2} - 2x^5 + \frac{x}{4} + 1\)[/tex]
2. [tex]\(-2x^5 + \frac{x^3}{2} + \frac{x}{4} + 1\)[/tex]
3. [tex]\(-2x^5 + \frac{x}{4} + \frac{x^3}{2} + 1\)[/tex]
4. [tex]\(1 - 2x^5 + \frac{x^3}{2} + \frac{x}{4}\)[/tex]
Let's break down the process step-by-step:
1. Identify the terms of the polynomial:
- The term with the highest degree is [tex]\(-2x^5\)[/tex].
- The next highest degree term is [tex]\(\frac{x^3}{2}\)[/tex].
- Then comes the term [tex]\(\frac{x}{4}\)[/tex].
- Finally, we have the constant term [tex]\(1\)[/tex].
2. Arrange these terms in descending order of their degrees:
- Start with the highest degree term: [tex]\(-2x^5\)[/tex].
- Follow this with the next highest degree term: [tex]\(\frac{x^3}{2}\)[/tex].
- Then place the next term: [tex]\(\frac{x}{4}\)[/tex].
- Finally, add the constant term [tex]\(1\)[/tex].
3. After ordering the terms, the polynomial in its standard form should look like:
[tex]\[
-2x^5 + \frac{x^3}{2} + \frac{x}{4} + 1
\][/tex]
Reviewing the given options, we see that option 2 matches this ordered form:
[tex]\(-2x^5 + \frac{x^3}{2} + \frac{x}{4} + 1\)[/tex].
Thus, the polynomial that represents the standard form of the original polynomial is:
[tex]\[
\boxed{-2x^5 + \frac{x^3}{2} + \frac{x}{4} + 1}
\][/tex]
