To combine the like terms and put the given polynomial [tex]\(\frac{1}{3} x+8 x^4-\frac{2}{3} x^2-x\)[/tex] in standard form, follow these steps:
1. Identify and group the like terms in the polynomial. Here, like terms are terms that have the same variable raised to the same power.
- [tex]\(\frac{1}{3} x\)[/tex] and [tex]\(-x\)[/tex] are like terms because they both have [tex]\(x\)[/tex] raised to the first power.
- The term [tex]\(8x^4\)[/tex] is already in its simplest form.
- The term [tex]\(-\frac{2}{3} x^2\)[/tex] is also already in its simplest form.
2. Combine the like terms:
- Combine [tex]\(\frac{1}{3} x\)[/tex] and [tex]\(-x\)[/tex]:
[tex]\[\frac{1}{3} x - x\][/tex]
To combine these, it's often helpful to rewrite [tex]\(-x\)[/tex] as a fraction with a common denominator:
[tex]\[\frac{1}{3} x - \frac{3}{3} x = \frac{1 - 3}{3} x = -\frac{2}{3} x\][/tex]
3. Write the polynomial in standard form. The standard form of a polynomial is written from the highest degree term to the lowest degree term:
[tex]\[8x^4 - \frac{2}{3} x^2 - \frac{2}{3} x\][/tex]
Therefore, the polynomial that correctly combines the like terms and puts the polynomial in standard form is:
[tex]\[8x^4 - \frac{2}{3} x^2 - \frac{2}{3} x\][/tex]
Among the choices given:
1. [tex]\(8 x^4-\frac{1}{3} x^2-x\)[/tex]
2. [tex]\(8 x^4+\frac{2}{3} x^2+\frac{2}{3} x\)[/tex]
3. [tex]\(8 x^4-\frac{2}{3} x^2-\frac{2}{3} x\)[/tex]
4. [tex]\(8 x^4-\frac{2}{3} x^2-x\)[/tex]
The correct answer is:
[tex]\[8 x^4-\frac{2}{3} x^2-\frac{2}{3} x\][/tex]
