To find the standard form of the given polynomial and correctly combine the like terms, let us first look at the original polynomial provided:
[tex]\[ \frac{1}{3}x + 8x^4 - \frac{2}{3}x^2 - x \][/tex]
We need to combine the like terms.
1. Identifying Like Terms:
- The term [tex]\(\frac{1}{3}x\)[/tex] and [tex]\(-x\)[/tex] are both first-degree terms (terms involving [tex]\(x\)[/tex]).
- The term [tex]\(-\frac{2}{3}x^2\)[/tex] is a second-degree term.
- The term [tex]\(8x^4\)[/tex] is a fourth-degree term.
2. Combining the First-Degree Terms:
- Combine [tex]\(\frac{1}{3}x\)[/tex] and [tex]\(-x\)[/tex].
[tex]\[ \frac{1}{3}x - x = \frac{1}{3}x - \frac{3}{3}x = -\frac{2}{3}x \][/tex]
3. Writing the Polynomial in Standard Form:
- After combining the like terms, the polynomial can be written in descending order of degree:
- The fourth-degree term is [tex]\(8x^4\)[/tex].
- The second-degree term is [tex]\(-\frac{2}{3}x^2\)[/tex].
- The first-degree term is [tex]\(-\frac{2}{3}x\)[/tex].
Thus, the polynomial in standard form is:
[tex]\[ 8x^4 - \frac{2}{3}x^2 - \frac{2}{3}x \][/tex]
Hence, the correct answer is:
[tex]\[ 8x^4 - \frac{2}{3}x^2 - \frac{2}{3}x \][/tex]
Therefore, the correct option among the given choices is:
[tex]\[ 8x^4 - \frac{2}{3}x^2 - \frac{2}{3}x \][/tex]
