To determine which terms can be added to the term [tex]\(3x^2y\)[/tex] to result in a monomial, we need to understand what a monomial is. A monomial is a single term consisting of a product of numbers and variables, where the variables can have non-negative integer exponents.
In algebra, terms are considered "like terms" if they have the same variables raised to the same powers. Only like terms can be added or subtracted to form another monomial.
The given term is [tex]\(3x^2y\)[/tex]. Therefore, any term that can combine with [tex]\(3x^2y\)[/tex] to form a monomial must also have the variables [tex]\(x^2y\)[/tex].
We will go through each provided term to check whether it can be combined with [tex]\(3x^2y\)[/tex]:
1. [tex]\(3xy\)[/tex]
- Variables: [tex]\(x\)[/tex] and [tex]\(y\)[/tex]
- Powers: [tex]\(x^1y^1\)[/tex]
- [tex]\(3xy\)[/tex] does not have the same variables and power as [tex]\(3x^2y\)[/tex].
2. [tex]\(-12x^2y\)[/tex]
- Variables: [tex]\(x^2\)[/tex] and [tex]\(y\)[/tex]
- Powers: [tex]\(x^2y^1\)[/tex]
- [tex]\(-12x^2y\)[/tex] has the same variables and powers as [tex]\(3x^2y\)[/tex] and is thus a like term.
3. [tex]\(2x^2y^2\)[/tex]
- Variables: [tex]\(x^2\)[/tex] and [tex]\(y^2\)[/tex]
- Powers: [tex]\(x^2y^2\)[/tex]
- [tex]\(2x^2y^2\)[/tex] does not have the same powers as [tex]\(3x^2y\)[/tex].
4. [tex]\(7xy^2\)[/tex]
- Variables: [tex]\(x\)[/tex] and [tex]\(y^2\)[/tex]
- Powers: [tex]\(x^1y^2\)[/tex]
- [tex]\(7xy^2\)[/tex] does not have the same variables and powers as [tex]\(3x^2y\)[/tex].
5. [tex]\(-10x^2\)[/tex]
- Variables: [tex]\(x^2\)[/tex]
- Powers: [tex]\(x^2y^0\)[/tex]
- [tex]\(-10x^2\)[/tex] does not have the same variables as [tex]\(3x^2y\)[/tex].
6. [tex]\(4x^2\)[/tex]
- Variables: [tex]\(x^2\)[/tex]
- Powers: [tex]\(x^2y^0\)[/tex]
- [tex]\(4x^2\)[/tex] does not have the same variables as [tex]\(3x^2y\)[/tex].
7. [tex]\(3x^3\)[/tex]
- Variables: [tex]\(x^3\)[/tex]
- Powers: [tex]\(x^3y^0\)[/tex]
- [tex]\(3x^3\)[/tex] does not have the same variables as [tex]\(3x^2y\)[/tex].
Only one term from the list can be added to [tex]\(3x^2y\)[/tex] to result in a monomial. That term is:
[tex]\[ -12x^2y \][/tex]
In summary, the only term that can be added to [tex]\(3x^2y\)[/tex] to result in a monomial is:
- [tex]\(-12x^2y\)[/tex]
