To determine which terms could be the missing first term of an expression that, when fully simplified, would be a binomial with a degree of 4, we need to understand the degree of each term provided. A binomial has two distinct terms, and the degree of the binomial is determined by the highest degree among its terms.
Let's analyze the degree of each provided term separately:
1. [tex]\(-5xy^3 + 9x^2y\)[/tex]
- The term [tex]\(-5xy^3\)[/tex] has a degree of [tex]\(1+3 = 4\)[/tex] (since the exponent of [tex]\(x\)[/tex] is 1, and the exponent of [tex]\(y\)[/tex] is 3).
- The term [tex]\(9x^2y\)[/tex] has a degree of [tex]\(2+1 = 3\)[/tex] (since the exponent of [tex]\(x\)[/tex] is 2, and the exponent of [tex]\(y\)[/tex] is 1).
2. [tex]\(0\)[/tex]
- This term is [tex]\(0\)[/tex] and does not contribute to the degree of a polynomial.
3. [tex]\(5xy^3\)[/tex]
- This term has a degree of [tex]\(1+3 = 4\)[/tex] (since the exponent of [tex]\(x\)[/tex] is 1, and the exponent of [tex]\(y\)[/tex] is 3).
4. [tex]\(9x^2y\)[/tex]
- This term has a degree of [tex]\(2+1 = 3\)[/tex] (since the exponent of [tex]\(x\)[/tex] is 2, and the exponent of [tex]\(y\)[/tex] is 1).
5. [tex]\(8.4\)[/tex]
- This is a constant term and has a degree of 0.
6. [tex]\(4xy^3\)[/tex]
- This term has a degree of [tex]\(1+3 = 4\)[/tex] (since the exponent of [tex]\(x\)[/tex] is 1, and the exponent of [tex]\(y\)[/tex] is 3).
After identifying the degrees of each term, we need to form a binomial with the highest degree of 4. Looking at the terms, [tex]\(-5xy^3\)[/tex], [tex]\(5xy^3\)[/tex], and [tex]\(4xy^3\)[/tex] all have a degree of 4.
Given that we need to choose three options that could serve as the missing first term in an expression that, when fully simplified, is a binomial with a degree of 4, we can choose any of these three terms as they already have a degree of 4 and can contribute to forming a binomial of degree 4 when paired with appropriate terms.
Therefore, the three options are:
- [tex]\( -5xy^3 + 9x^2y \)[/tex]
- [tex]\( 5xy^3 \)[/tex]
- [tex]\( 4xy^3 \)[/tex]
