To write a polynomial in standard form, the terms must be ordered in descending powers of the variable [tex]\( x \)[/tex]. Let's take the given polynomial:
[tex]\[ \frac{x}{4} - 2x^5 + \frac{x^3}{2} + 1 \][/tex]
First, identify and list the individual terms along with their degrees:
- [tex]\( -2x^5 \)[/tex] : degree 5
- [tex]\( \frac{x^3}{2} \)[/tex] : degree 3
- [tex]\( \frac{x}{4} \)[/tex] : degree 1
- [tex]\( 1 \)[/tex] : degree 0
The standard form polynomial should arrange these terms from highest to lowest degree:
1. Start with the term of the highest degree, which is [tex]\( -2x^5 \)[/tex].
2. Next, include the term with the next highest degree, which is [tex]\( \frac{x^3}{2} \)[/tex].
3. Then, include the term with the next lower degree, which is [tex]\( \frac{x}{4} \)[/tex].
4. Finally, include the constant term, which is [tex]\( 1 \)[/tex].
So, we should combine these terms as follows:
[tex]\[ -2x^5 + \frac{x^3}{2} + \frac{x}{4} + 1 \][/tex]
To convert this to a common denominator and combine into a single rational expression:
[tex]\[
\frac{x}{4} - 2x^5 + \frac{x^3}{2} + 1 = \frac{-8x^5 + 2x^3 + x + 4}{4}
\][/tex]
Thus, the polynomial in standard form is:
[tex]\[
\boxed{\frac{-8x^5 + 2x^3 + x + 4}{4}}
\][/tex]
Among the given options, the polynomial that matches this standard form is:
[tex]\[ -2 x^5 + \frac{x^3}{2} + \frac{x}{4} + 1 \][/tex]
So the correct answer is:
[tex]\[ -2 x^5 + \frac{x^3}{2} + \frac{x}{4} + 1 \][/tex]
