Question

which term could be put in the blank to create a fully simplified polynomial written in standard form?
$8x^3y^2 - \underline{quadquad} + 3xy^2 - 4y^3$
$\circ\\ x^2y^2$
$\circ\\ x^3y^3$
$\circ\\ 7xy^2$
$\circ\\ 7x^3y^3$

Explanation:

Step1: Recall standard polynomial rules

A fully simplified standard polynomial has no like terms, and terms are ordered by descending degree (sum of exponents of variables in each term).

Step2: Calculate term degrees

• Degree of $8x^3y^2$: $3+2=5$
• Degree of $3xy^2$: $1+2=3$
• Degree of $-4y^3$: $3$

Step3: Analyze each option

• Option 1 ($x^2y^2$): Degree $4$, no like terms in the given polynomial. When inserted, the polynomial becomes $8x^3y^2 - x^2y^2 + 3xy^2 - 4y^3$, which is simplified and in standard order (degree 5 → 4 → 3 → 3).
• Option 2 ($x^3y^3$): Degree $6$, which would come before $8x^3y^2$ in standard form, breaking the order.
• Option 3 ($7xy^2$): Like term with $3xy^2$, so combining them would be required, meaning the polynomial would not be fully simplified as given.
• Option 4 ($7x^3y^3$): Degree $6$, which would come before $8x^3y^2$ in standard form, breaking the order.

Answer:

A. $x^2y^2$