To determine the polynomial written in standard form, we must organize the given terms by the degree of [tex]\(x\)[/tex] from highest to lowest. The degree of a term is determined by the sum of the exponents of the variables in that term.
Let's start by identifying the degrees of each term in the polynomials provided:
1. [tex]\(8 x^2 y^2\)[/tex]: The sum of exponents is [tex]\(2 + 2 = 4\)[/tex].
2. [tex]\(-3 x^3 y\)[/tex]: The sum of exponents is [tex]\(3 + 1 = 4\)[/tex].
3. [tex]\(4 x^4\)[/tex]: The sum of exponents is [tex]\(4\)[/tex].
4. [tex]\(-7 x y^3\)[/tex]: The sum of exponents is [tex]\(1 + 3 = 4\)[/tex].
Since all terms have the same degree, we can focus on organizing them consistently in the following way: First, by the highest degree of [tex]\(x\)[/tex] and second, by any remaining variables consistently.
Now let's determine the order for each polynomial provided:
1. [tex]\(8 x^2 y^2-3 x^3 y+4 x^4-7 x y^3\)[/tex]
- Ordering by degrees: [tex]\(4 x^4\)[/tex], [tex]\(-3 x^3 y\)[/tex], [tex]\(8 x^2 y^2\)[/tex], [tex]\(-7 x y^3\)[/tex]
2. [tex]\(4 x^4-3 x^3 y+8 x^2 y^2-7 x y^3\)[/tex]
- This polynomial is correctly ordered: [tex]\(4 x^4\)[/tex], [tex]\(-3 x^3 y\)[/tex], [tex]\(8 x^2 y^2\)[/tex], [tex]\(-7 x y^3\)[/tex]
3. [tex]\(4 x^4-7 x y^3-3 x^3 y+8 x^2 y^2\)[/tex]
- The order is not consistent.
4. [tex]\(4 x^4+8 x^2 y^2-3 x^3 y-7 x y^3\)[/tex]
- The order is not correct since [tex]\(8 x^2 y^2\)[/tex] appears before [tex]\(-3 x^3 y\)[/tex].
5. [tex]\(-7 x y^3-3 x^3 y+8 x^2 y^2+4 x^4\)[/tex]
- The order is not correct since it starts with [tex]\(-7 x y^3\)[/tex] instead of [tex]\(4 x^4\)[/tex].
Examining all the options, it is clear that the polynomial [tex]\(4 x^4-3 x^3 y+8 x^2 y^2-7 x y^3\)[/tex] is in standard form with terms ordered by the degree of [tex]\(x\)[/tex] from highest to lowest and then by systematically by any remaining variables.
Therefore, the polynomial in standard form is:
[tex]\[ \boxed{4 x^4-3 x^3 y+8 x^2 y^2-7 x y^3} \][/tex]
