To find which term should be the first term in Alina's polynomial when written in standard form, let's follow the step-by-step process of simplifying and combining like terms.
Here is the polynomial given:
[tex]\[ x y^2 - 2 x^2 y + 3 y^3 - 6 x^2 y + 4 x y^2 \][/tex]
### Step 1: Identify and Combine Like Terms
The polynomial consists of different terms that we can group together based on their variable components.
1. [tex]\( x y^2 \)[/tex] terms:
- [tex]\( x y^2 \)[/tex]
- [tex]\( 4 x y^2 \)[/tex]
2. [tex]\( x^2 y \)[/tex] terms:
- [tex]\( -2 x^2 y \)[/tex]
- [tex]\( -6 x^2 y \)[/tex]
3. A single [tex]\( y^3 \)[/tex] term:
- [tex]\( 3 y^3 \)[/tex]
### Step 2: Combine the Like Terms
For [tex]\( x y^2 \)[/tex] terms:
[tex]\[ x y^2 + 4 x y^2 = 5 x y^2 \][/tex]
For [tex]\( x^2 y \)[/tex] terms:
[tex]\[ -2 x^2 y - 6 x^2 y = -8 x^2 y \][/tex]
The [tex]\( y^3 \)[/tex] term remains as is:
[tex]\[ 3 y^3 \][/tex]
Thus, after combining like terms, the polynomial is:
[tex]\[ 5 x y^2 - 8 x^2 y + 3 y^3 \][/tex]
### Step 3: Write the Polynomial in Standard Form
In standard form, a polynomial is written in descending order of the exponents of the variables. The term [tex]\( 3 y^3 \)[/tex] is stated to be the last term. Thus:
[tex]\[ 3 y^3 \][/tex]
First, let's determine the highest-order term among [tex]\( 5 x y^2 \)[/tex] and [tex]\( -8 x^2 y \)[/tex]:
- [tex]\( x^2 y \)[/tex] has a higher degree (sum of exponents is 3: [tex]\( 2 + 1 \)[/tex]).
- [tex]\( x y^2 \)[/tex] has a lower degree (sum of exponents 3: [tex]\( 1 + 2 \)[/tex]).
When comparing by traditional priority, prioritize the term with higher degree in one variable, and in this case, the order of terms favor [tex]\( x^2 \)[/tex] terms.
So we arrange the terms in standard form:
[tex]\[ -8 x^2 y + 5 x y^2 + 3 y^3 \][/tex]
Thus, the first term in standard form is:
[tex]\[
\boxed{-8 x^2 y}
\][/tex]
