To determine which terms could be used as the first term of the expression to create a polynomial with a degree of 5 in standard form, we need to calculate the total degree of each term. The degree of each term is determined by adding the exponents of the variables in that term.
Let's calculate the degree for each term:
1. [tex]\(2x^2y^2 - 3xy^3\)[/tex]:
- For [tex]\(2x^2y^2\)[/tex]:
[tex]\[
\text{Degree} = \text{Exponent of } x + \text{Exponent of } y = 2 + 2 = 4
\][/tex]
- For [tex]\(-3xy^3\)[/tex]:
[tex]\[
\text{Degree} = \text{Exponent of } x + \text{Exponent of } y = 1 + 3 = 4
\][/tex]
- Therefore, the highest degree in the term [tex]\(2x^2y^2 - 3xy^3\)[/tex] is 4.
2. [tex]\(-4x^3y^2\)[/tex]:
[tex]\[
\text{Degree} = \text{Exponent of } x + \text{Exponent of } y = 3 + 2 = 5
\][/tex]
3. [tex]\(x^3\)[/tex]:
[tex]\[
\text{Degree} = \text{Exponent of } x = 3
\][/tex]
4. [tex]\(8.4x^4y^2\)[/tex]:
[tex]\[
\text{Degree} = \text{Exponent of } x + \text{Exponent of } y = 4 + 2 = 6
\][/tex]
5. [tex]\(5x^4y\)[/tex]:
[tex]\[
\text{Degree} = \text{Exponent of } x + \text{Exponent of } y = 4 + 1 = 5
\][/tex]
6. [tex]\(-xay^3\)[/tex]:
[tex]\[
\text{Degree} = \text{Exponent of } x + \text{Exponent of } a + \text{Exponent of } y = 1 + 1 + 3 = 5 \text{ (since a is also considered a variable here)}
\][/tex]
7. [tex]\(\frac{-2x^4}{y} \rightarrow -2x^4y^{-1}\)[/tex]:
[tex]\[
\text{Degree} = \text{Exponent of } x + \text{Exponent of } \frac{1}{y} = 4 - 1 = 3
\][/tex]
Let's identify the terms with a degree of 5:
- [tex]\(-4x^3y^2\)[/tex] (Degree = 5)
- [tex]\(5x^4y\)[/tex] (Degree = 5)
- [tex]\(-xay^3\)[/tex] (Degree = 5)
Therefore, the terms that could be the first term of the expression to create a polynomial with a degree of 5 in standard form are:
- [tex]\(-4x^3y^2\)[/tex]
- [tex]\(5x^4y\)[/tex]
- [tex]\(-xay^3\)[/tex]
