To find the polynomial that represents the standard form of the original polynomial [tex]\(\frac{x}{4}-2 x^5+\frac{x^3}{2}+1\)[/tex], we need to rearrange the terms in descending order of the exponents of [tex]\(x\)[/tex].
Here is the step-by-step process:
1. Identify the terms and their respective exponents:
- [tex]\(-2x^5\)[/tex] has an exponent of 5.
- [tex]\(\frac{x^3}{2}\)[/tex] has an exponent of 3.
- [tex]\(\frac{x}{4}\)[/tex] has an exponent of 1.
- [tex]\(1\)[/tex] is a constant term with an exponent of 0.
2. Rearrange the terms in descending order:
- The highest exponent is 5, so we start with [tex]\(-2x^5\)[/tex].
- The next highest exponent is 3, which gives us [tex]\(\frac{x^3}{2}\)[/tex].
- Then, we have the term with an exponent of 1, [tex]\(\frac{x}{4}\)[/tex].
- Finally, we add the constant term [tex]\(1\)[/tex].
3. Construct the polynomial in standard form:
- Arranging the terms in order, we get [tex]\(-2x^5 + \frac{x^3}{2} + \frac{x}{4} + 1\)[/tex].
So, the polynomial that represents the standard form of the given polynomial [tex]\(\frac{x}{4}-2 x^5+\frac{x^3}{2}+1\)[/tex] is:
[tex]\[-2 x^5+\frac{x^3}{2}+\frac{x}{4}+1.\][/tex]
Hence, the correct choice is:
[tex]\[
-2 x^5+\frac{x^3}{2}+\frac{x}{4}+1
\][/tex]
This corresponds to the second option in the given choices. Therefore, the correct answer is:
[tex]\[
2
\][/tex]
