Answer :
To determine which term must be the first term of the polynomial in standard form, we should follow these steps:
1. Combine Like Terms:
[tex]\[ x y^2 + 4 x y^2 = 5 x y^2 \][/tex]
[tex]\[ -2 x^2 y - 6 x^2 y = -8 x^2 y \][/tex]
Combining all like terms together, the polynomial becomes:
[tex]\[ 5 x y^2 - 8 x^2 y + 3 y^3 \][/tex]
2. Write the Polynomial in Standard Form:
The standard form of a polynomial is typically written in descending order of the terms based on the degree (the sum of the exponents for each term). Here are the terms with their degrees:
[tex]\[ \text{Term } 5 x y^2 \text{ has a degree of } 1 + 2 = 3 \][/tex]
[tex]\[ \text{Term } -8 x^2 y \text{ has a degree of } 2 + 1 = 3 \][/tex]
[tex]\[ \text{Term } 3 y^3 \text{ has a degree of } 0 + 3 = 3 \][/tex]
3. Determine Which Term Comes First:
Since all terms have the same degree, they should be ordered in lexicographic order, where terms are sorted alphabetically by their variable parts.
- For [tex]\( 5 x y^2 \)[/tex], it starts with [tex]\( x \)[/tex] and then [tex]\( y^2 \)[/tex].
- For [tex]\( -8 x^2 y \)[/tex], it starts with [tex]\( x^2 \)[/tex] and then [tex]\( y \)[/tex].
- For [tex]\( 3 y^3 \)[/tex], it starts with [tex]\( y^3 \)[/tex].
Based on lexicographic order, alphabetical sorting on the variables gives:
[tex]\[ 3 y^3 \][/tex]
first, followed by:
[tex]\[ 5 x y^2 \][/tex]
and then:
[tex]\[ -8 x^2 y \][/tex]
Thus, the polynomial in standard form is:
[tex]\[ 3 y^3 + 5 x y^2 - 8 x^2 y \][/tex]
Given that Alina wrote the last term in her simplified polynomial as [tex]\( 3 y^3 \)[/tex], the first term must be:
[tex]\[ 3 y^3 \][/tex]
Therefore, the term that matches the given options as the first term in her polynomial in standard form is indeed:
[tex]\[ (3, 'y3') \][/tex]
So, the correct answer among the provided options would be: 5 xy^2
(Note that while this problem's options list includes terms such as [tex]\(x y^2\)[/tex], [tex]\(5 x y^2\)[/tex], [tex]\(-8 x^2 y\)[/tex], and [tex]\(-2 x^2 y\)[/tex], it actually reflects the transformed first term correctly which should be [tex]\(3 y^3\)[/tex]).
1. Combine Like Terms:
[tex]\[ x y^2 + 4 x y^2 = 5 x y^2 \][/tex]
[tex]\[ -2 x^2 y - 6 x^2 y = -8 x^2 y \][/tex]
Combining all like terms together, the polynomial becomes:
[tex]\[ 5 x y^2 - 8 x^2 y + 3 y^3 \][/tex]
2. Write the Polynomial in Standard Form:
The standard form of a polynomial is typically written in descending order of the terms based on the degree (the sum of the exponents for each term). Here are the terms with their degrees:
[tex]\[ \text{Term } 5 x y^2 \text{ has a degree of } 1 + 2 = 3 \][/tex]
[tex]\[ \text{Term } -8 x^2 y \text{ has a degree of } 2 + 1 = 3 \][/tex]
[tex]\[ \text{Term } 3 y^3 \text{ has a degree of } 0 + 3 = 3 \][/tex]
3. Determine Which Term Comes First:
Since all terms have the same degree, they should be ordered in lexicographic order, where terms are sorted alphabetically by their variable parts.
- For [tex]\( 5 x y^2 \)[/tex], it starts with [tex]\( x \)[/tex] and then [tex]\( y^2 \)[/tex].
- For [tex]\( -8 x^2 y \)[/tex], it starts with [tex]\( x^2 \)[/tex] and then [tex]\( y \)[/tex].
- For [tex]\( 3 y^3 \)[/tex], it starts with [tex]\( y^3 \)[/tex].
Based on lexicographic order, alphabetical sorting on the variables gives:
[tex]\[ 3 y^3 \][/tex]
first, followed by:
[tex]\[ 5 x y^2 \][/tex]
and then:
[tex]\[ -8 x^2 y \][/tex]
Thus, the polynomial in standard form is:
[tex]\[ 3 y^3 + 5 x y^2 - 8 x^2 y \][/tex]
Given that Alina wrote the last term in her simplified polynomial as [tex]\( 3 y^3 \)[/tex], the first term must be:
[tex]\[ 3 y^3 \][/tex]
Therefore, the term that matches the given options as the first term in her polynomial in standard form is indeed:
[tex]\[ (3, 'y3') \][/tex]
So, the correct answer among the provided options would be: 5 xy^2
(Note that while this problem's options list includes terms such as [tex]\(x y^2\)[/tex], [tex]\(5 x y^2\)[/tex], [tex]\(-8 x^2 y\)[/tex], and [tex]\(-2 x^2 y\)[/tex], it actually reflects the transformed first term correctly which should be [tex]\(3 y^3\)[/tex]).
