To determine which polynomial correctly combines the like terms and expresses the given polynomial in standard form, we can follow these steps:
1. Identify and group like terms: The given polynomial is:
[tex]\[
9 x y^3 - 4 y^4 - 10 x^2 y^2 + x^3 y + 3 x^4 + 2 x^2 y^2 - 9 y^4
\][/tex]
2. Combine the coefficients of the like terms:
- For [tex]\(x^4\)[/tex]: There is only one term [tex]\(3 x^4\)[/tex].
- For [tex]\(x^3 y\)[/tex]: There is only one term [tex]\(x^3 y\)[/tex].
- For [tex]\(x^2 y^2\)[/tex]: Combine [tex]\(-10 x^2 y^2\)[/tex] and [tex]\(2 x^2 y^2\)[/tex]:
[tex]\[
-10 + 2 = -8 \quad \text{thus,} \quad -8 x^2 y^2
\][/tex]
- For [tex]\(x y^3\)[/tex]: There is only one term [tex]\(9 x y^3\)[/tex].
- For [tex]\(y^4\)[/tex]: Combine [tex]\(-4 y^4\)[/tex] and [tex]\(-9 y^4\)[/tex]:
[tex]\[
-4 - 9 = -13 \quad \text{thus,} \quad -13 y^4
\][/tex]
3. Arrange the polynomial in standard form: The standard form arranges terms in order of decreasing powers:
So the simplified and combined polynomial is:
[tex]\[
3 x^4 + x^3 y - 8 x^2 y^2 + 9 x y^3 - 13 y^4
\][/tex]
Hence, the polynomial that correctly combines the like terms and expresses the given polynomial in standard form is:
[tex]$
3 x^4 + x^3 y - 8 x^2 y^2 + 9 x y^3 - 13 y^4
$[/tex]
Comparing this with the given options, the correct choice is:
[tex]$
3 x^4 + x^3 y - 8 x^2 y^2 + 9 x y^3 - 13 y^4
$[/tex]
