To determine which student is correct about the standard form of the polynomial expression, we need to recall the conventions for writing polynomials.
1. Polynomial Structure:
- Polynomials should be written in standard form, typically with the terms ordered by degree.
- For multivariable polynomials, we tend to prioritize variables based on convention, usually in alphabetical order or any given problem's specific guideline. Here we'll consider the standard approach which prioritizes [tex]\( x \)[/tex] over [tex]\( y \)[/tex].
2. Expressions Provided:
- Marcus’s expression: [tex]\( 3x^3 - 4x^2 y + y^3 + 2 \)[/tex]
- Ariel’s expression: [tex]\( y^3 - 4x^2 y + 3x^3 + 2 \)[/tex]
3. Simplifying and Comparing:
- Both expressions contain the same terms and can be expanded as follows:
[tex]\[ 3x^3 - 4x^2 y + y^3 + 2 \][/tex]
[tex]\[ y^3 - 4x^2 y + 3x^3 + 2 \][/tex]
- Each term in the expressions is identical; they just differ in the order in which the terms are written.
4. Standard Form Conventions:
- In standard form, terms should be written in descending order of powers. For multivariable polynomials prioritizing the variable [tex]\( x \)[/tex] first:
- We first list the terms with the highest powers of [tex]\( x \)[/tex] and among same power of [tex]\( x \)[/tex], we can sort by the next variable [tex]\( y \)[/tex].
- In Marcus’s polynomial: [tex]\( 3x^3 \)[/tex] has the highest power of [tex]\( x \)[/tex], followed by [tex]\( -4x^2 y \)[/tex] (which has [tex]\( x^2 \)[/tex] and [tex]\( y \)[/tex]), then [tex]\( y^3 \)[/tex] (as it lacks [tex]\( x \)[/tex]), and finally the constant term [tex]\( +2 \)[/tex].
5. Conclusion:
- Marcus’s polynomial, [tex]\( 3x^3 - 4x^2 y + y^3 + 2 \)[/tex], is in standard form because the terms are correctly ordered in descending powers of [tex]\( x \)[/tex].
- Ariel’s polynomial, [tex]\( y^3 - 4x^2 y + 3x^3 + 2 \)[/tex], places [tex]\( y^3 \)[/tex] before [tex]\( 3x^3 \)[/tex], which does not follow the convention of ordering by the highest powers of [tex]\( x \)[/tex] first.
Therefore, Marcus is correct in stating that the polynomial expression [tex]\( 3x^3 - 4x^2 y + y^3 + 2 \)[/tex] is in standard form.
