To write a polynomial in standard form, we need to arrange its terms in descending order according to their degrees. The degree of a term is indicated by the highest power of the variable [tex]\( x \)[/tex] in that term.
Given polynomial:
[tex]\[
\frac{x^2}{2} - 3x + 4x^3 + 6
\][/tex]
Let's identify the degree of each term:
- [tex]\(4x^3\)[/tex] has a degree of 3.
- [tex]\(\frac{x^2}{2}\)[/tex] has a degree of 2.
- [tex]\(-3x\)[/tex] has a degree of 1.
- [tex]\(6\)[/tex] is a constant term with a degree of 0.
Now, we order these terms from the highest degree to the lowest degree:
1. The term with degree 3: [tex]\( 4x^3 \)[/tex]
2. The term with degree 2: [tex]\(\frac{x^2}{2}\)[/tex]
3. The term with degree 1: [tex]\(-3x\)[/tex]
4. The constant term (degree 0): [tex]\( 6 \)[/tex]
Therefore, the polynomial in standard form is:
[tex]\[
4x^3 + \frac{x^2}{2} - 3x + 6
\][/tex]
Now, let's match this expression with the options given:
1. [tex]\(6 + \frac{x^2}{2} - 3x + 4x^3\)[/tex] is not in the correct order.
2. [tex]\(4x^3 + \frac{x^2}{2} - 3x + 6\)[/tex] matches our polynomial in standard form.
3. [tex]\(-3x + 4x^3 + 6 + \frac{x^2}{2}\)[/tex] is not in the correct order.
4. [tex]\(\frac{x^2}{2} + 4x^3 - 3x + 6\)[/tex] is not in the correct order.
The correct choice is:
[tex]\[
\boxed{4x^3 + \frac{x^2}{2} - 3x + 6}
\][/tex]
