To determine the standard form of the polynomial [tex]\(9 x^2 y-4 x+3 y^3 x-2 y^2\)[/tex], we need to arrange the polynomial in descending order of the variables and their respective powers. Here's how you can approach the task step by step:
1. Identify all the terms:
The given polynomial has the following terms:
[tex]\[
9 x^2 y, -4 x, 3 y^3 x, -2 y^2
\][/tex]
2. Classify each term by the combined degree of variables (considering each variable individually and in combination):
- [tex]\(9 x^2 y\)[/tex]: The combined degree is [tex]\(2 + 1 = 3\)[/tex].
- [tex]\(-4 x\)[/tex]: The combined degree is [tex]\(1\)[/tex] (since it’s only in terms of [tex]\(x\)[/tex], no [tex]\(y\)[/tex]).
- [tex]\(3 y^3 x\)[/tex]: The combined degree is [tex]\(3 + 1 = 4\)[/tex].
- [tex]\(-2 y^2\)[/tex]: The combined degree is [tex]\(2\)[/tex] (since it’s only in terms of [tex]\(y\)[/tex], no [tex]\(x\)[/tex]).
3. Arrange the terms in descending order based on the combined degree:
- Highest combined degree first: [tex]\(3 y^3 x\)[/tex] (combined degree = 4)
- Next highest combined degree: [tex]\(9 x^2 y\)[/tex] (combined degree = 3)
- Next term: [tex]\(-2 y^2\)[/tex] (combined degree = 2)
- Last: [tex]\(-4 x\)[/tex] (combined degree = 1)
4. Write the polynomial in descending order:
After organizing the terms, the standard form is:
[tex]\[
3 y^3 x + 9 x^2 y - 2 y^2 - 4 x
\][/tex]
5. Match this with the provided options:
- Option 1: [tex]\(9 x^2 y-4 x+3 y^3 x-2 y^2\)[/tex]
- Option 2: [tex]\(3 y^3 x-2 y^2+9 x^2 y-4 x\)[/tex]
- Option 3: [tex]\(9 x^2 y-4 x-2 y^2+3 y^3 x\)[/tex]
- Option 4: [tex]\(3 y^3 x+9 x^2 y-2 y^2-4 x\)[/tex]
From this analysis, the correct match for the standard form of [tex]\(9 x^2 y-4 x+3 y^3 x-2 y^2\)[/tex] is:
[tex]\[\boxed{4}\][/tex]
