To determine which term can be added to the polynomial [tex]\( 8x^3 y^2 + 3x y^2 - 4 y^3 \)[/tex] in order to create a fully simplified polynomial in standard form, let’s analyze each option provided.
1. Option [tex]\( x^2 y^2 \)[/tex]
- When we consider the term [tex]\( x^2 y^2 \)[/tex], there are no like terms in the polynomial [tex]\( 8x^3 y^2 + 3x y^2 - 4 y^3 \)[/tex]. This term cannot combine with any of the existing terms to simplify the polynomial.
2. Option [tex]\( x^3 y^3 \)[/tex]
- The term [tex]\( x^3 y^3 \)[/tex] does not align with any terms in the given polynomial either. Therefore, it similarly does not contribute to simplifying the polynomial.
3. Option [tex]\( 7x y^2 \)[/tex]
- The term [tex]\( 7x y^2 \)[/tex] can combine with the term [tex]\( 3x y^2 \)[/tex] from the polynomial:
[tex]\[
3x y^2 + 7x y^2 = 10x y^2
\][/tex]
- By adding [tex]\( 7x y^2 \)[/tex] to the polynomial, we get:
[tex]\[
8x^3 y^2 + 3x y^2 + 7x y^2 - 4 y^3 = 8x^3 y^2 + 10x y^2 - 4 y^3
\][/tex]
- This is a proper combination that simplifies [tex]\( 3x y^2 \)[/tex] and [tex]\( 7x y^2 \)[/tex].
4. Option [tex]\( 7 x^0 y^3 \)[/tex]
- First, recognize that [tex]\( 7 x^0 y^3 \)[/tex] is equivalent to [tex]\( 7 y^3 \)[/tex] (since [tex]\( x^0 \)[/tex] is 1).
- The term [tex]\( 7 y^3 \)[/tex] can be combined with the term [tex]\( -4 y^3 \)[/tex] from the polynomial:
[tex]\[
-4 y^3 + 7 y^3 = 3 y^3
\][/tex]
- By adding [tex]\( 7 y^3 \)[/tex] to the polynomial, we get:
[tex]\[
8x^3 y^2 + 3x y^2 - 4 y^3 + 7 y^3 = 8x^3 y^2 + 3x y^2 + 3 y^3
\][/tex]
- This combination is also correct as it simplifies [tex]\( -4 y^3 \)[/tex] and [tex]\( 7 y^3 \)[/tex].
To create a polynomial written in standard form, typically higher degree terms are written first followed by terms of lower degree, combining like terms as much as possible. Both option 3 and option 4 provide a correct way to simplify the polynomial. However, considering typical conventions for arranging terms in standard form, we aim to have the highest degree term listed.
Given these considerations, the term that could be best put in the blank to fully simplify the polynomial and maintain the polynomial in a consistent and decreasing degree order is:
[tex]\[
\boxed{7 x y^2}
\][/tex]
