To determine which terms can be part of an expression that forms a binomial of degree 4 when combined with [tex]\(9x^2y\)[/tex], we need to analyze the degrees of each option when added to [tex]\(9x^2y\)[/tex].
### Step-by-Step Solution
1. Understand the Degree of a Polynomial Term:
The degree of a term in a polynomial is the sum of the exponents of its variables. For example, the term [tex]\(9x^2y\)[/tex] has a degree of [tex]\(2 \text{ (from } x^2) + 1 \text{ (from } y) = 3\)[/tex].
2. List the Given Options:
These are the options provided:
- [tex]\(-5xy^3\)[/tex]
- [tex]\(5xy^3\)[/tex]
- [tex]\(9x^2y\)[/tex]
- [tex]\(8y^4\)[/tex]
- [tex]\(4xy^3\)[/tex]
3. Calculate the Degree of Each Option:
- Option 1: [tex]\(-5xy^3\)[/tex]
- Degree: [tex]\(1 \text{ (from } x) + 3 \text{ (from } y^3) = 1 + 3 = 4\)[/tex]
- Option 2: [tex]\(5xy^3\)[/tex]
- Degree: [tex]\(1 \text{ (from } x) + 3 \text{ (from } y^3) = 1 + 3 = 4\)[/tex]
- Option 3: [tex]\(9x^2y\)[/tex]
- Degree: [tex]\(2 \text{ (from } x^2) + 1 \text{ (from } y) = 2 + 1 = 3\)[/tex]
- Option 4: [tex]\(8y^4\)[/tex]
- Degree: [tex]\(0 \text{ (from } x) + 4 \text{ (from } y^4) = 0 + 4 = 4\)[/tex]
- Option 5: [tex]\(4xy^3\)[/tex]
- Degree: [tex]\(1 \text{ (from } x) + 3 \text{ (from } y^3) = 1 + 3 = 4\)[/tex]
4. Determine Valid Options:
We are looking for terms such that, when added to [tex]\(9x^2y\)[/tex], the resulting expression is a binomial of degree 4. Any term with a degree of 4 is valid.
- [tex]\(-5xy^3\)[/tex]: degree 4 (valid)
- [tex]\(5xy^3\)[/tex]: degree 4 (valid)
- [tex]\(8y^4\)[/tex]: degree 4 (valid)
- [tex]\(4xy^3\)[/tex]: degree 4 (valid)
5. Conclusion:
The terms that could be the missing part of the expression to make it a binomial with a degree of 4 are:
- [tex]\(-5xy^3\)[/tex]
- [tex]\(5xy^3\)[/tex]
- [tex]\(8y^4\)[/tex]
- [tex]\(4xy^3\)[/tex]
Since we need to select three options, we have:
[tex]\[
\boxed{-5xy^3, \; 5xy^3, \; 8y^4, \; 4xy^3}
\][/tex]
