To determine which polynomial correctly combines the like terms and expresses the given polynomial \( 9xy^3 - 4y^4 - 10x^2y^2 + x^3y + 3x^4 + 2x^2y^2 - 9y^4 \) in standard form, follow these detailed steps:
1. Identify and Combine Like Terms:
The given polynomial contains several terms. We need to combine the like terms:
- \( -4y^4 \) and \( -9y^4 \) are both of the form \( y^4 \).
- \( -10x^2y^2 \) and \( 2x^2y^2 \) are both of the form \( x^2y^2 \).
Combine them:
- \( -4y^4 - 9y^4 = -13y^4 \)
- \( -10x^2y^2 + 2x^2y^2 = -8x^2y^2 \)
2. List the Remaining Terms:
The remaining terms that do not have like terms are:
- \( 3x^4 \)
- \( x^3y \)
- \( 9xy^3 \)
3. Arrange in Standard Form:
The standard form of a polynomial arranges the terms in descending order by the degree of \( x \) and \( y \):
Thus, the combined and simplified polynomial is:
[tex]\[
3x^4 - 8x^2y^2 + x^3y + 9xy^3 - 13y^4
\][/tex]
Therefore, the polynomial correctly combining like terms and expressed in standard form is:
[tex]\[
3x^4 - 8x^2y^2 + x^3y + 9xy^3 - 13y^4
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{3 x^4 - 8 x^2 y^2 + x^3 y + 9 x y^3 - 13 y^4}
\][/tex]
