To determine which polynomial is in standard form, we need to check the degree of each term in each polynomial and see if the terms are arranged in descending order of their degrees. The degree of a term is the sum of the exponents of its variables.
Let's analyze each polynomial step-by-step.
### Polynomial 1: [tex]\(2 x y^3+3 x^3 y^4-4 x^4 y^5+9 x^5 y^5\)[/tex]
1. [tex]\(2 x y^3\)[/tex]:
- Degree: [tex]\(1 + 3 = 4\)[/tex]
2. [tex]\(3 x^3 y^4\)[/tex]:
- Degree: [tex]\(3 + 4 = 7\)[/tex]
3. [tex]\(-4 x^4 y^5\)[/tex]:
- Degree: [tex]\(4 + 5 = 9\)[/tex]
4. [tex]\(9 x^5 y^5\)[/tex]:
- Degree: [tex]\(5 + 5 = 10\)[/tex]
In descending order of degree: 10, 9, 7, 4. The polynomial is already in standard form.
### Polynomial 2: [tex]\(8 x^5-5 y^4-2 x y^5+x^2 y\)[/tex]
1. [tex]\(8 x^5\)[/tex]:
- Degree: [tex]\(5\)[/tex]
2. [tex]\(-5 y^4\)[/tex]:
- Degree: [tex]\(4\)[/tex]
3. [tex]\(-2 x y^5\)[/tex]:
- Degree: [tex]\(1 + 5 = 6\)[/tex]
4. [tex]\(x^2 y\)[/tex]:
- Degree: [tex]\(2 + 1 = 3\)[/tex]
Degrees: 6, 5, 4, 3. The polynomial is not in descending order of degree.
### Polynomial 3: [tex]\(x^3 y^2-3 x^2 y-9 x^4 y^3+11 x\)[/tex]
1. [tex]\(x^3 y^2\)[/tex]:
- Degree: [tex]\(3 + 2 = 5\)[/tex]
2. [tex]\(-3 x^2 y\)[/tex]:
- Degree: [tex]\(2 + 1 = 3\)[/tex]
3. [tex]\(-9 x^4 y^3\)[/tex]:
- Degree: [tex]\(4 + 3 = 7\)[/tex]
4. [tex]\(11 x\)[/tex]:
- Degree: [tex]\(1\)[/tex]
Degrees: 7, 5, 3, 1. The polynomial is not in descending order of degree.
### Polynomial 4: [tex]\(5 x^7 y^2+7 x^6 y^5-14 x^3 y^7+51 x^2 y^9\)[/tex]
1. [tex]\(5 x^7 y^2\)[/tex]:
- Degree: [tex]\(7 + 2 = 9\)[/tex]
2. [tex]\(7 x^6 y^5\)[/tex]:
- Degree: [tex]\(6 + 5 = 11\)[/tex]
3. [tex]\(-14 x^3 y^7\)[/tex]:
- Degree: [tex]\(3 + 7 = 10\)[/tex]
4. [tex]\(51 x^2 y^9\)[/tex]:
- Degree: [tex]\(2 + 9 = 11\)[/tex]
Degrees: 11, 11, 10, 9. The polynomial is already in descending order of degree.
## Conclusion
Both Polynomial 1 and Polynomial 4 appear to be in standard form, as their terms are arranged in descending order of degree.
However, typically when only one correct answer is required, the first occurring correct polynomial is chosen.
Therefore, the correct answer is Polynomial 1:
[tex]\[ 2 x y^3 + 3 x^3 y^4 - 4 x^4 y^5 + 9 x^5 y^5 \][/tex] is in standard form.
