Question

Marcus states that the polynomial expression 3x3 – 4x2y + y3 + 2 is in standard form. Ariel states that it should be y3 – 4x2y + 3x3 +2. Explain which student is correct and why.

Answer 1

Both expressions are algebraically identical, so neither is "wrong" as a polynomial. Which one is called "standard form" depends on the ordering convention you adopt.

Common conventions:
- For single-variable polynomials: "standard form" means terms in descending powers of the variable (highest degree first).
- For multivariable polynomials: a usual convention is to order by total degree (highest total degree first) and, among equal-degree terms, use a fixed variable order (often x before y) or a lexicographic order.

In your polynomial the nonconstant terms all have total degree 3, so a tie-breaker is needed. If you use the common convention x > y (or lex order with x first), the order is
3x^3 − 4x^2y + y^3 + 2,
so Marcus’s ordering matches the usual convention and would be called standard form. Ariel’s ordering would be "standard" only if you had chosen the opposite variable order (y > x) or a different ordering rule.