To find the appropriate term to fill in the blank in the polynomial \(8x^3y^2 - \_ + 3xy^2 - 4y^3\) such that it is fully simplified and written in standard form, let's carefully analyze the given options and align them with the structure of the polynomial.
The standard form of a polynomial arranges the terms in descending order of their degrees (where the degree is the sum of the exponents of the variables).
Given terms:
1. \(8x^3y^2\): Degree = \(3 + 2 = 5\)
2. \(3xy^2\): Degree = \(1 + 2 = 3\)
3. \(-4y^3\): Degree = \(0 + 3 = 3\)
We need the missing term to appropriately fit within the polynomial while maintaining the descending order of degrees. Let's evaluate each option:
1. \(x^2y^2\): Degree = \(2 + 2 = 4\)
- This term has a lower total degree than \(8x^3y^2\) but higher than \(3xy^2\). While it maintains the order, it's not an ideal fit for standard form support.
2. \(x^3y^3\): Degree = \(3 + 3 = 6\)
- This term has a higher degree than \(8x^3y^2\), which would disrupt the descending order if placed after \(8x^3y^2\). Thus, it does not fit.
3. \(7xy^2\): Degree = \(1 + 2 = 3\)
- This term maintains the proper degree sequence since immediately after the highest \(8x^3y^2\) degree term, a fitting term here would facilitate an ordered reduction. Also note, this coefficient fits polynomial styling and combines seamlessly.
4. \(7x^9y^3\): Degree = \(9 + 3 = 12\)
- This term has a far higher degree than the given terms, making it unsuited for this polynomial.
Upon review of the degrees and fitting within the polynomial style and standard expectations:
The term that can be correctly inserted into the blank to maintain the polynomial's proper descending order and fully simplify it is:
[tex]\[ 7xy^2 \][/tex]
Thus, the fully simplified polynomial in standard form would be:
[tex]\[ 8x^3y^2 - 7xy^2 + 3xy^2 - 4y^3 \][/tex]
