To determine which polynomial is in standard form, we need to look at each polynomial and see if the terms are written in descending order of their degrees.
### Explanation of Degree of a Term:
The degree of a term in a polynomial is the sum of the exponents of the variables in that term. For example, in the term [tex]\(6x^3y^2\)[/tex]:
- The degree is [tex]\(3 + 2 = 5\)[/tex].
In the polynomial, the terms should be arranged in descending order based on their degrees.
### Breakdown of Each Polynomial:
1. First Polynomial:
[tex]\[3xy + 6x^3y^2 - 4x^4y^3 + 19x^7y^4\][/tex]
- Degrees of terms:
- [tex]\(3xy\)[/tex] has degree [tex]\(1 + 1 = 2\)[/tex]
- [tex]\(6x^3y^2\)[/tex] has degree [tex]\(3 + 2 = 5\)[/tex]
- [tex]\(4x^4y^3\)[/tex] has degree [tex]\(4 + 3 = 7\)[/tex]
- [tex]\(19x^7y^4\)[/tex] has degree [tex]\(7 + 4 = 11\)[/tex]
The terms are in ascending order of their degrees, not descending order.
2. Second Polynomial:
[tex]\[18x^5 - 7x^2y - 2xy^2 + 17y^4\][/tex]
- Degrees of terms:
- [tex]\(18x^5\)[/tex] has degree [tex]\(5\)[/tex]
- [tex]\(-7x^2y\)[/tex] has degree [tex]\(2 + 1 = 3\)[/tex]
- [tex]\(-2xy^2\)[/tex] has degree [tex]\(1 + 2 = 3\)[/tex]
- [tex]\(17y^4\)[/tex] has degree [tex]\(4\)[/tex]
The terms are not in descending order of their degrees.
3. Third Polynomial:
[tex]\[x^5y^5 - 3xy - 11x^2y^2 + 12\][/tex]
- Degrees of terms:
- [tex]\(x^5y^5\)[/tex] has degree [tex]\(5 + 5 = 10\)[/tex]
- [tex]\(-3xy\)[/tex] has degree [tex]\(1 + 1 = 2\)[/tex]
- [tex]\(-11x^2y^2\)[/tex] has degree [tex]\(2 + 2 = 4\)[/tex]
- [tex]\(12\)[/tex] has degree [tex]\(0\)[/tex]
The terms are not in descending order of their degrees.
4. Fourth Polynomial:
[tex]\[15 + 12xy^2 - 11x^9y^5 + 5x^7y^2\][/tex]
- Degrees of terms:
- [tex]\(15\)[/tex] has degree [tex]\(0\)[/tex]
- [tex]\(12xy^2\)[/tex] has degree [tex]\(1 + 2 = 3\)[/tex]
- [tex]\(-11x^9y^5\)[/tex] has degree [tex]\(9 + 5 = 14\)[/tex]
- [tex]\(5x^7y^2\)[/tex] has degree [tex]\(7 + 2 = 9\)[/tex]
The terms are in descending order of their degrees: [tex]\(14, 9, 3, 0\)[/tex].
### Conclusion:
The fourth polynomial is the one that is in standard form, with terms ordered in descending order based on their degrees:
[tex]\[15 + 12xy^2 - 11x^9y^5 + 5x^7y^2\][/tex]
Therefore, the polynomial in standard form is the fourth one.
### Final Answer:
The polynomial in standard form is:
[tex]\[ \boxed{4} \][/tex]
