Question

explaining standard form for multivaria
marcus states that the polynomial expression $3x^3 - 4x^2y + y^3 + 2$ is in standard form. ariel states that it should be $y^3 - 4x^2y + 3x^3 + 2$. explain which student is correct and why.

Explanation:

Standard form for a multivariable polynomial is ordered by descending total degree of each term, and if terms have equal total degree, they can be ordered by descending degree of one variable (either $x$ or $y$). First, calculate the total degree of each term:

• $3x^3$: total degree $3$
• $-4x^2y$: total degree $2+1=3$
• $y^3$: total degree $3$
• $2$: total degree $0$

All terms of degree 3 can be ordered by descending degree of $x$ (Marcus's version: $3x^3, -4x^2y, y^3$) or descending degree of $y$ (Ariel's version: $y^3, -4x^2y, 3x^3$). Both orderings are valid for standard form as long as terms are grouped by descending total degree, and the constant term is last.

Answer:

Both students are correct. Standard form for a multivariable polynomial requires terms to be ordered by descending total degree, which both expressions follow (all degree-3 terms come before the degree-0 constant term). For terms with the same total degree, ordering can be done by descending degree of either $x$ (Marcus's approach) or $y$ (Ariel's approach), and both are acceptable conventions for standard form.