Let's examine each term to determine whether adding it to [tex]\(3x^2y\)[/tex] will result in a new valid monomial.
A monomial is a single term expression consisting of a product of constants and variables, where the variables have non-negative integer exponents. For the given monomial [tex]\(3x^2y\)[/tex], any term that, when added, maintains this form will be considered.
### Given Term
The given term is:
[tex]\[ 3x^2y \][/tex]
### Potential Terms to Add
Let's analyze each potential term one by one:
1. [tex]\(3xy\)[/tex]
- Adding [tex]\(3xy\)[/tex] would result in a polynomial [tex]\(3x^2y + 3xy\)[/tex], not a monomial.
- Not valid.
2. [tex]\(-12x^2y\)[/tex]
- Adding [tex]\(-12x^2y\)[/tex] would result in:
[tex]\[ 3x^2y - 12x^2y = -9x^2y \][/tex]
- The result [tex]\(-9x^2y\)[/tex] is still a monomial.
- Valid.
3. [tex]\(2x^2y^2\)[/tex]
- Adding [tex]\(2x^2y^2\)[/tex] would result in:
[tex]\[ 3x^2y + 2x^2y^2 \][/tex]
- The result is not a single term but a polynomial with different powers of [tex]\(y\)[/tex].
- Not valid.
4. [tex]\(7xy^2\)[/tex]
- Adding [tex]\(7xy^2\)[/tex] would result in:
[tex]\[ 3x^2y + 7xy^2 \][/tex]
- The result is not a single term but a polynomial with different powers of [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
- Not valid.
5. [tex]\(-10x^2\)[/tex]
- Adding [tex]\(-10x^2\)[/tex] would result in:
[tex]\[ 3x^2y - 10x^2\][/tex]
- The result is not a single term but a polynomial with different powers of [tex]\(x\)[/tex].
- Not valid.
6. [tex]\(4x^2y\)[/tex]
- Adding [tex]\(4x^2y\)[/tex] would result in:
[tex]\[ 3x^2y + 4x^2y = 7x^2y \][/tex]
- The result [tex]\(7x^2y\)[/tex] is still a monomial.
- Valid.
7. [tex]\(3x^3\)[/tex]
- Adding [tex]\(3x^3\)[/tex] would result in:
[tex]\[ 3x^2y + 3x^3\][/tex]
- The result is not a single term but a polynomial with different powers of [tex]\(x\)[/tex].
- Not valid.
### Valid Terms
The terms that would result in a monomial when added to [tex]\(3x^2y\)[/tex] are:
- [tex]\(-12x^2y\)[/tex]
- [tex]\(4x^2y\)[/tex]
Therefore, the valid terms are:
[tex]\[
-12x^2y, \quad 4x^2y
\][/tex]
