To find the correct polynomial that combines like terms and expresses the given polynomial in standard form, we should combine the like terms step-by-step. Let's break down the expression:
Given polynomial:
[tex]\[ 9xy^3 - 4y^4 - 10x^2y^2 + x^3y + 3x^4 + 2x^2y^2 - 9y^4 \][/tex]
1. Combine the [tex]\( y^4 \)[/tex] terms:
[tex]\[ -4y^4 - 9y^4 = -13y^4 \][/tex]
2. Combine the [tex]\( x^2y^2 \)[/tex] terms:
[tex]\[ -10x^2y^2 + 2x^2y^2 = -8x^2y^2 \][/tex]
3. There are no other like terms to combine, so let's write the polynomial combining the results:
[tex]\[ 3x^4 + x^3y + 9xy^3 - 13y^4 - 8x^2y^2 \][/tex]
Rewriting in decreasing order of degrees, we get:
[tex]\[ 3x^4 + x^3y - 8x^2y^2 + 9xy^3 - 13y^4 \][/tex]
We now need to compare this with the given options to see which polynomial matches this form.
The options are:
1. [tex]\( 9xy^3 - 4y^4 - 10x^2y^2 + x^3y + 3x^4 + 2x^2y^2 - 9y^4 \)[/tex]
2. [tex]\( -13y^4 + 3x^4 - 8x^2y^2 + x^3y + 9xy^3 \)[/tex]
3. [tex]\( -13y^4 + x^3y - 8x^2y^2 + 9xy^3 + 3x^4 \)[/tex]
4. [tex]\( 3x^4 - 8x^2y^2 + x^3y + 9xy^3 - 13y^4 \)[/tex]
5. [tex]\( 3x^4 + x^3y - 8x^2y^2 + 9xy^3 - 13y^4 \)[/tex]
The polynomial that matches our combined expression is indeed:
[tex]\[ 3x^4 + x^3y - 8x^2y^2 + 9xy^3 - 13y^4 \][/tex]
Therefore, the corresponding option is:
[tex]\[ 3x^4 + x^3y - 8x^2y^2 + 9xy^3 - 13y^4 \][/tex]
Thus, the correct polynomial is [tex]\( 3x^4 + x^3y - 8x^2y^2 + 9xy^3 - 13y^4 \)[/tex]. This matches the fifth option.
