To determine which term could be put in the blank to create a fully simplified polynomial written in standard form, we need to place the terms in descending order based on the total degree of each term. Recall that the degree of a term in a polynomial is the sum of the exponents of all the variables in that term.
Given polynomial: [tex]\(8x^3y^2 - \_\_\_\_ + 3xy^2 - 4y^3\)[/tex]
Let's analyze the degrees of each term:
1. [tex]\(8x^3y^2\)[/tex] has a degree of [tex]\(3 + 2 = 5\)[/tex]
2. [tex]\(3xy^2\)[/tex] has a degree of [tex]\(1 + 2 = 3\)[/tex]
3. [tex]\(-4y^3\)[/tex] has a degree of [tex]\(0 + 3 = 3\)[/tex]
Now let's check the given options for the missing term in the polynomial and determine their degrees:
1. [tex]\(x^2y^2\)[/tex] has a degree of [tex]\(2 + 2 = 4\)[/tex]
2. [tex]\(x^3y^3\)[/tex] has a degree of [tex]\(3 + 3 = 6\)[/tex]
3. [tex]\(7xy^2\)[/tex] has a degree of [tex]\(1 + 2 = 3\)[/tex]
4. [tex]\(7x^0y^3\)[/tex] simplifies to [tex]\(7y^3\)[/tex] and has a degree of [tex]\(0 + 3 = 3\)[/tex]
Now we need to find a term whose degree fits in descending order between the degrees of the existing terms.
The degrees order we need is: 5, ___, 3, 3
Analyzing the options again:
- [tex]\(x^2y^2\)[/tex] with degree 4 fits between 5 and 3.
- [tex]\(x^3y^3\)[/tex] with degree 6 is greater than 5, so it does not fit.
- [tex]\(7xy^2\)[/tex] with degree 3 is not between 5 and 3, it matches existing degrees.
- [tex]\(7x^0y^3\)[/tex] or [tex]\(7y^3\)[/tex] with degree 3 is also not between 5 and 3, it matches existing degrees.
Thus, the term that fits the polynomial in standard descending order is [tex]\(x^2y^2\)[/tex].
Therefore, the term that could be put in the blank is:
[tex]\[ x^2 y^2 \][/tex]
The correct answer is:
[tex]\[ 1 \][/tex]
