To determine which terms can be added to [tex]\(3x^2y\)[/tex] to result in a monomial, we need to understand the definition of a monomial. A monomial is a single term algebraic expression consisting of a coefficient and variables raised to non-negative integer powers, with each variable having only one unique power.
Given the term [tex]\(3x^2y\)[/tex], any term that can be added to it to form a monomial must have the same variable components [tex]\(x^2\)[/tex] and [tex]\(y\)[/tex] combined with their respective coefficients.
Let's evaluate each provided term against [tex]\(3x^2y\)[/tex]:
1. [tex]\(3xy\)[/tex]:
- Variables: [tex]\(x\)[/tex] and [tex]\(y\)[/tex]
- Coefficients: 3
- Powers: [tex]\(x^1\)[/tex], [tex]\(y^1\)[/tex]
The powers of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] do not match [tex]\(x^2\)[/tex] and [tex]\(y\)[/tex], hence this term is not compatible.
2. [tex]\(-12x^2y\)[/tex]:
- Variables: [tex]\(x\)[/tex], [tex]\(y\)[/tex]
- Coefficients: -12
- Powers: [tex]\(x^2\)[/tex], [tex]\(y\)[/tex]
The variables and their powers match exactly with [tex]\(x^2\)[/tex] and [tex]\(y\)[/tex] in [tex]\(3x^2y\)[/tex]. Adding this term would result in:
[tex]\[
3x^2y + (-12x^2y) = (3 - 12)x^2y = -9x^2y
\][/tex]
which is a monomial. This term is compatible.
3. [tex]\(2x^2y^2\)[/tex]:
- Variables: [tex]\(x\)[/tex], [tex]\(y\)[/tex]
- Coefficients: 2
- Powers: [tex]\(x^2\)[/tex], [tex]\(y^2\)[/tex]
The power of [tex]\(y\)[/tex] does not match. Hence, this term is not compatible.
4. [tex]\(7xy^2\)[/tex]:
- Variables: [tex]\(x\)[/tex], [tex]\(y\)[/tex]
- Coefficients: 7
- Powers: [tex]\(x\)[/tex], [tex]\(y^2\)[/tex]
The powers do not match as [tex]\(x\)[/tex] does not have the suitable power, and [tex]\(y\)[/tex] also has a different power. Hence, this term is not compatible.
5. [tex]\(-10x^2\)[/tex]:
- Variables: [tex]\(x\)[/tex]
- Coefficients: -10
- Powers: [tex]\(x^2\)[/tex]
This term is missing [tex]\(y\)[/tex]. Hence, this term is not compatible.
6. [tex]\(4x^2y\)[/tex]:
- Variables: [tex]\(x\)[/tex], [tex]\(y\)[/tex]
- Coefficients: 4
- Powers: [tex]\(x^2\)[/tex], [tex]\(y\)[/tex]
The variables and the powers match exactly. Adding this term would result in:
[tex]\[
3x^2y + 4x^2y = (3 + 4)x^2y = 7x^2y
\][/tex]
which is a monomial. This term is compatible.
7. [tex]\(3x^3\)[/tex]:
- Variables: [tex]\(x\)[/tex]
- Coefficients: 3
- Powers: [tex]\(x^3\)[/tex]
The power of [tex]\(x\)[/tex] does not match. Hence, this term is not compatible.
Therefore, the terms that can be added to [tex]\(3x^2y\)[/tex] resulting in a monomial are:
[tex]\[
-12x^2y, \quad 4x^2y
\][/tex]
