Certainly! Let's analyze the polynomial expressions provided by Marcus and Ariel to determine which one follows the standard form for polynomials.
### Understanding Polynomial Standard Form
For a polynomial in multiple variables, there are conventions for how to order the terms:
1. Order by Powers of One Variable First: Typically, terms are ordered primarily by the descending powers of one variable.
2. Resolve Ties by Secondary Variables: If the powers of the primary variable are the same, we then order by the descending powers of another variable.
### Analyzing Marcus's Expression
Marcus's polynomial is:
[tex]\[ 3x^3 - 4x^2y + y^3 + 2 \][/tex]
### Analyzing Ariel's Expression
Ariel's polynomial is:
[tex]\[ y^3 - 4x^2y + 3x^3 + 2 \][/tex]
### Compare Both Expressions by Conventional Ordering
1. Marcus's Ordering:
[tex]\[ 3x^3 - 4x^2y + y^3 + 2 \][/tex]
- Terms in Marcus's version are:
- [tex]\( 3x^3 \)[/tex]
- [tex]\( -4x^2y \)[/tex]
- [tex]\( y^3 \)[/tex]
- [tex]\( +2 \)[/tex]
- Ordered by descending powers of [tex]\( x \)[/tex] first:
- [tex]\( 3x^3 \)[/tex]
- [tex]\( -4x^2y \)[/tex]
- [tex]\( y^3 \)[/tex]
- [tex]\( +2 \)[/tex]
2. Ariel's Ordering:
[tex]\[ y^3 - 4x^2y + 3x^3 + 2 \][/tex]
- Terms in Ariel's version are:
- [tex]\( y^3 \)[/tex]
- [tex]\( -4x^2y \)[/tex]
- [tex]\( 3x^3 \)[/tex]
- [tex]\( +2 \)[/tex]
- Ordered by descending powers of [tex]\( y \)[/tex] first:
- [tex]\( y^3 \)[/tex]
- [tex]\( -4x^2y \)[/tex]
- [tex]\( 3x^3 \)[/tex]
- [tex]\( +2 \)[/tex]
### Conventional Standard Form
The conventional way to write polynomials, especially in contexts where one variable is typically given precedence, is to order by the descending powers of [tex]\( x \)[/tex] first.
### Conclusion
Given that the conventional ordering is generally by descending powers of [tex]\( x \)[/tex] first, Marcus's ordering of the polynomial:
[tex]\[ 3x^3 - 4x^2y + y^3 + 2 \][/tex]
is the correct way to write the polynomial in standard form.
Hence, Marcus is correct because he ordered the polynomial in descending powers of [tex]\( x \)[/tex], which aligns with the common convention for writing polynomials.
