To determine the first term of Alina's polynomial in standard form, we must analyze the given choices based on polynomial hierarchy rules.
First, let's recall that the standard form of a polynomial lists its terms in descending order of their degrees. The degree of a term is the sum of the exponents of the variables in that term.
The last term in Alina's polynomial is [tex]\(3y^3\)[/tex]. This term has a degree of 3, as the exponent of [tex]\(y\)[/tex] is 3.
Next, we need to examine the degrees of each of the given options to identify the highest-degree term, which should come first if all terms are of equal degree:
1. [tex]\(xy^2\)[/tex] has a degree of [tex]\(1 + 2 = 3\)[/tex], because the exponent of [tex]\(x\)[/tex] is 1 and the exponent of [tex]\(y\)[/tex] is 2.
2. [tex]\(5xy^2\)[/tex] has a degree of [tex]\(1 + 2 = 3\)[/tex], using similar reasoning as above.
3. [tex]\(-8x^2y\)[/tex] has a degree of [tex]\(2 + 1 = 3\)[/tex], because the exponent of [tex]\(x\)[/tex] is 2 and the exponent of [tex]\(y\)[/tex] is 1.
4. [tex]\(-2x^2y\)[/tex] has a degree of [tex]\(2 + 1 = 3\)[/tex], following the same approach as in the previous term.
Now, we observe that all given terms have the same degree of 3. When terms have the same degree, we typically list them in order based on the exponents of the variables. In this case, we start by looking at the exponents of [tex]\(x\)[/tex] since [tex]\(x\)[/tex] comes before [tex]\(y\)[/tex] alphabetically:
- [tex]\(-8x^2y\)[/tex] and [tex]\(-2x^2y\)[/tex] have [tex]\(x^2\)[/tex] terms.
- [tex]\(xy^2\)[/tex] and [tex]\(5xy^2\)[/tex] have [tex]\(x^1\)[/tex] terms.
Since terms involving [tex]\(x^2\)[/tex] should come before those with [tex]\(x^1\)[/tex], we need to choose between [tex]\(-8x^2y\)[/tex] and [tex]\(-2x^2y\)[/tex].
Between [tex]\(-8x^2y\)[/tex] and [tex]\(-2x^2y\)[/tex], we choose the one that appears first in conventional ordering or polynomial notation. Generally, coefficients do not influence the order, but if we adhere to typical notation, selecting the higher magnitude coefficient term first is common.
Therefore, the first term in Alina's polynomial in standard form is:
[tex]\[-8x^2y.\][/tex]
Thus, the answer is [tex]\(-8 x^2 y\)[/tex].
