To determine which polynomial is in its standard form, we need to examine the degrees of the terms in each polynomial and ensure that they are arranged in descending order.
### 1. [tex]\(3xy + 6x^3y^2 - 4x^4y^3 + 19x^7y^4\)[/tex]
Let's determine the degree of each term:
- [tex]\(3xy\)[/tex]: The degree is [tex]\(1 + 1 = 2\)[/tex].
- [tex]\(6x^3y^2\)[/tex]: The degree is [tex]\(3 + 2 = 5\)[/tex].
- [tex]\(4x^4y^3\)[/tex]: The degree is [tex]\(4 + 3 = 7\)[/tex].
- [tex]\(19x^7y^4\)[/tex]: The degree is [tex]\(7 + 4 = 11\)[/tex].
The terms are arranged in ascending order of degree: [tex]\(2, 5, 7, 11\)[/tex]. Therefore, this polynomial is not in standard form.
### 2. [tex]\(18x^5 - 7x^2y - 2xy^2 + 17y^4\)[/tex]
Determine the degree of each term:
- [tex]\(18x^5\)[/tex]: The degree is [tex]\(5 + 0 = 5\)[/tex].
- [tex]\(-7x^2y\)[/tex]: The degree is [tex]\(2 + 1 = 3\)[/tex].
- [tex]\(-2xy^2\)[/tex]: The degree is [tex]\(1 + 2 = 3\)[/tex].
- [tex]\(17y^4\)[/tex]: The degree is [tex]\(0 + 4 = 4\)[/tex].
The terms should be arranged in descending order of degrees: [tex]\(5, 4, 3, 3\)[/tex]. However, the second and third term also need to be compared separately. The polynomial is partially in descending order but not fully standardized as the terms [tex]\(3\)[/tex] should be placed together in the order of alphabetically assigned (x then y).
### 3. [tex]\(x^5y^5 - 3xy - 11x^2y^2 + 12\)[/tex]
Determine the degree of each term:
- [tex]\(x^5y^5\)[/tex]: The degree is [tex]\(5 + 5 = 10\)[/tex].
- [tex]\(-3xy\)[/tex]: The degree is [tex]\(1 + 1 = 2\)[/tex].
- [tex]\(-11x^2y^2\)[/tex]: The degree is [tex]\(2 + 2 = 4\)[/tex].
- [tex]\(12\)[/tex]: The degree is [tex]\(0 + 0 = 0\)[/tex].
The degrees [tex]\(10, 4, 2, 0\)[/tex] are in descending order. Therefore, this polynomial is in standard form.
### 4. [tex]\(15 + 12xy^2 - 11x^9y^5 + 5x^7y^2\)[/tex]
Determine the degree of each term:
- [tex]\(15\)[/tex]: The degree is [tex]\(0 + 0 = 0\)[/tex].
- [tex]\(12xy^2\)[/tex]: The degree is [tex]\(1 + 2 = 3\)[/tex].
- [tex]\(-11x^9y^5\)[/tex]: The degree is [tex]\(9 + 5 = 14\)[/tex].
- [tex]\(5x^7y^2\)[/tex]: The degree is [tex]\(7 + 2 = 9\)[/tex].
The degrees [tex]\(0, 3, 14, 9\)[/tex] are not in descending order. Therefore, this polynomial is not in standard form.
Based on our analysis, the polynomial in standard form is:
[tex]\[ x^5y^5 - 3xy - 11x^2y^2 + 12 \][/tex]
