To correctly combine the like terms of the given polynomial and express it in standard form, let's follow these steps:
1. Identify the like terms:
The given polynomial is:
[tex]\[
9xy^3 - 4y^4 - 10x^2y^2 + x^3y + 3x^4 + 2x^2y^2 - 9y^4
\][/tex]
- Like terms involving [tex]\( y^4 \)[/tex]:
[tex]\( -4y^4 \)[/tex] and [tex]\( -9y^4 \)[/tex]
- Like terms involving [tex]\( x^2y^2 \)[/tex]:
[tex]\( -10x^2y^2 \)[/tex] and [tex]\( 2x^2y^2 \)[/tex]
- Terms that do not have like terms:
[tex]\( 9xy^3 \)[/tex], [tex]\( x^3y \)[/tex], and [tex]\( 3x^4 \)[/tex]
2. Combine the like terms:
- Combining the coefficients of [tex]\( y^4 \terms\)[/tex]:
[tex]\[
-4y^4 - 9y^4 = -13y^4
\][/tex]
- Combining the coefficients of [tex]\( x^2y^2 \)[/tex]:
[tex]\[
-10x^2y^2 + 2x^2y^2 = -8x^2y^2
\][/tex]
The terms [tex]\( 9xy^3 \)[/tex], [tex]\( x^3y \)[/tex], and [tex]\( 3x^4 \)[/tex] remain unchanged as they do not have like terms.
3. Construct the simplified polynomial in standard form:
The simplified polynomial is:
[tex]\[
-13y^4 + 3x^4 - 8x^2y^2 + x^3y + 9xy^3
\][/tex]
Now, let's match this to the provided choices:
[tex]\[
\begin{align*}
\text{a. } & -13 y^4+3 x^4-8 x^2 y^2+x^3 y+9 x y^3 \\
\text{b. } & -13 y^4+x^3 y-8 x^2 y^2+9 x y^3+3 x^4 \\
\text{c. } & 3 x^4-8 x^2 y^2+x^3 y+9 x y^3-13 y^4 \\
\text{d. } & 3 x^4+x^3 y-8 x^2 y^2+9 x y^3-13 y^4
\end{align*}
\][/tex]
Options a and b have the correct signs and coefficients:
- a. [tex]\( -13 y^4 + 3 x^4 - 8 x^2 y^2 + x^3 y + 9 x y^3 \)[/tex]
- b. [tex]\( -13 y^4 + x^3 y - 8 x^2 y^2 + 9 x y^3 + 3 x^4 \)[/tex]
Since standard form typically arranges terms in descending powers of variables, option a ([tex]\( -13 y^4 + 3 x^4 - 8 x^2 y^2 + x^3 y + 9 x y^3 \)[/tex]) is more appropriate.
Thus, the correct polynomial is:
[tex]\[
-13 y^4+3 x^4-8 x^2 y^2+x^3 y+9 x y^3
\][/tex]
