To determine which polynomial is in standard form, we need to identify the polynomial that is organized in descending order of the powers of the variables. Let's analyze each polynomial step-by-step.
1. Polynomial 1: [tex]\(3 x y + 6 x^3 y^2 - 4 x^4 y^3 + 19 x^7 y^4\)[/tex]
For each term, we can calculate the total power of [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
- [tex]\(3 x y\)[/tex] has [tex]\(1+1 = 2\)[/tex]
- [tex]\(6 x^3 y^2\)[/tex] has [tex]\(3+2 = 5\)[/tex]
- [tex]\(-4 x^4 y^3\)[/tex] has [tex]\(4+3 = 7\)[/tex]
- [tex]\(19 x^7 y^4\)[/tex] has [tex]\(7+4 = 11\)[/tex]
The terms written in descending order of total power are:
[tex]\[ 19 x^7 y^4, -4 x^4 y^3, 6 x^3 y^2, 3 x y \][/tex]
However, it does not exactly adhere to standard form when considering individual variable powers.
2. Polynomial 2: [tex]\(18 x^5 - 7 x^2 y - 2 x y^2 + 17 y^4\)[/tex]
The order for each term descending:
- [tex]\(18 x^5\)[/tex]
- [tex]\(-7 x^2 y\)[/tex]
- [tex]\(-2 x y^2\)[/tex]
- [tex]\(17 y^4\)[/tex]
These terms appear to be ordered correctly in terms of descending power combinations:
[tex]\[18 x^5, -7 x^2 y, -2 x y^2, 17 y^4\][/tex]
3. Polynomial 3: [tex]\(x^5 y^5 - 3 x y - 11 x^2 y^2 + 12\)[/tex]
For each term in descending order:
- [tex]\(x^5 y^5\)[/tex] has [tex]\(5+5 = 10\)[/tex]
- [tex]\(-3 x y\)[/tex] has [tex]\(1+1 = 2\)[/tex]
- [tex]\(-11 x^2 y^2\)[/tex] has [tex]\(2+2 = 4\)[/tex]
- [tex]\(12\)[/tex] has the constant power 0
Therefore, we have:
[tex]\[x^5 y^5, -11 x^2 y^2, -3 x y, 12\][/tex]
The combination powers are inconsistent when adding the constant part.
4. Polynomial 4: [tex]\(15 + 12 x y^2 - 11 x^9 y^5 + 5 x^7 y^2\)[/tex]
In descending order:
- [tex]\(15\)[/tex] has the constant 0
- [tex]\(12 x y^2\)[/tex] has [tex]\(1+2 = 3\)[/tex]
- [tex]\(5 x^7 y^2\)[/tex] has [tex]\(7+2 = 9\)[/tex]
- [tex]\(-11 x^9 y^5\)[/tex] has [tex]\(9+5 = 14\)[/tex]
If we order these by total variable powers, they would return:
[tex]\[-11 x^9 y^5, 5 x^7 y^2, 12 x y^2, 15 \][/tex]
After reviewing, the polynomial that properly orders the terms by descending powers of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] is:
[tex]\[ 18 x^5 - 7 x^2 y - 2 x y^2 + 17 y^4 \][/tex]
Thus, the polynomial in standard form is:
[tex]\[ \boxed{2} \][/tex]
