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Which polynomial correctly combines the like terms and puts the given polynomial in standard form?

[tex]\[ \frac{1}{3}x + 8x^4 - \frac{2}{3}x^2 - x \][/tex]

A. [tex]\( 8x^4 - \frac{1}{3}x^2 - x \)[/tex]
B. [tex]\( 8x^4 + \frac{2}{3}x^2 + \frac{2}{3}x \)[/tex]
C. [tex]\( 8x^4 - \frac{2}{3}x^2 - \frac{2}{3}x \)[/tex]
D. [tex]\( 8x^4 - \frac{2}{3}x^2 - x \)[/tex]


Sagot :

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]