Answer :
To determine which terms result in a monomial when added to [tex]\(3x^2y\)[/tex], we need to ensure that the resulting expression is a single term with matching variable parts and exponents.
Consider the given terms to be added to [tex]\(3x^2y\)[/tex]:
1. [tex]\(3xy\)[/tex]
2. [tex]\(-12x^2y\)[/tex]
3. [tex]\(2x^2y^2\)[/tex]
4. [tex]\(7xy^2\)[/tex]
5. [tex]\(-10x^2\)[/tex]
6. [tex]\(4x^2y\)[/tex]
7. [tex]\(3x^3\)[/tex]
### Step-by-Step Analysis:
1. [tex]\(3xy\)[/tex]:
- The term [tex]\(3xy\)[/tex] has the variables [tex]\(x\)[/tex] and [tex]\(y\)[/tex], but the exponents do not match [tex]\(3x^2y\)[/tex] (which should have [tex]\(x^2\)[/tex] and [tex]\(y\)[/tex]).
- Result after addition: [tex]\(3x^2y + 3xy\)[/tex]
- This results into a polynomial, not a monomial, because the terms cannot be combined into a single term due to differing exponents.
2. [tex]\(-12x^2y\)[/tex]:
- The term [tex]\(-12x^2y\)[/tex] has the exact same variables [tex]\(x^2\)[/tex] and [tex]\(y\)[/tex] as [tex]\(3x^2y\)[/tex].
- Result after addition: [tex]\(3x^2y - 12x^2y = -9x^2y\)[/tex]
- This is a single term (monomial).
3. [tex]\(2x^2y^2\)[/tex]:
- The term [tex]\(2x^2y^2\)[/tex] has [tex]\(x^2\)[/tex] and [tex]\(y^2\)[/tex], not matching [tex]\(3x^2y\)[/tex] (which only has [tex]\(y\)[/tex]).
- Result after addition: [tex]\(3x^2y + 2x^2y^2\)[/tex]
- This results into a polynomial, not a monomial, because the terms cannot be combined into a single term due to differing exponents.
4. [tex]\(7xy^2\)[/tex]:
- The term [tex]\(7xy^2\)[/tex] has the variables [tex]\(x\)[/tex] and [tex]\(y^2\)[/tex], not matching [tex]\(3x^2y\)[/tex] (which should have [tex]\(x^2\)[/tex]).
- Result after addition: [tex]\(3x^2y + 7xy^2\)[/tex]
- This results into a polynomial, not a monomial, because the terms cannot be combined into a single term due to differing exponents.
5. [tex]\(-10x^2\)[/tex]:
- The term [tex]\(-10x^2\)[/tex] lacks the variable [tex]\(y\)[/tex] entirely.
- Result after addition: [tex]\(3x^2y - 10x^2\)[/tex]
- This results into a polynomial, not a monomial, because the terms cannot be combined into a single term due to differing variables.
6. [tex]\(4x^2y\)[/tex]:
- The term [tex]\(4x^2y\)[/tex] has the exact same variables [tex]\(x^2\)[/tex] and [tex]\(y\)[/tex] as [tex]\(3x^2y\)[/tex].
- Result after addition: [tex]\(3x^2y + 4x^2y = 7x^2y\)[/tex]
- This is a single term (monomial).
7. [tex]\(3x^3\)[/tex]:
- The term [tex]\(3x^3\)[/tex] has [tex]\(x^3\)[/tex] instead of [tex]\(x^2\)[/tex], not matching [tex]\(3x^2y\)[/tex].
- Result after addition: [tex]\(3x^2y + 3x^3\)[/tex]
- This results into a polynomial, not a monomial, because the terms cannot be combined into a single term due to differing exponents.
### Conclusion:
The terms that would result in a monomial when added to [tex]\(3x^2y\)[/tex] are:
- [tex]\(-12x^2y\)[/tex]
- [tex]\(4x^2y\)[/tex]
So, the correct answers are:
- [tex]\(-12 x^2 y\)[/tex]
- [tex]\(4 x^2 y\)[/tex]
Consider the given terms to be added to [tex]\(3x^2y\)[/tex]:
1. [tex]\(3xy\)[/tex]
2. [tex]\(-12x^2y\)[/tex]
3. [tex]\(2x^2y^2\)[/tex]
4. [tex]\(7xy^2\)[/tex]
5. [tex]\(-10x^2\)[/tex]
6. [tex]\(4x^2y\)[/tex]
7. [tex]\(3x^3\)[/tex]
### Step-by-Step Analysis:
1. [tex]\(3xy\)[/tex]:
- The term [tex]\(3xy\)[/tex] has the variables [tex]\(x\)[/tex] and [tex]\(y\)[/tex], but the exponents do not match [tex]\(3x^2y\)[/tex] (which should have [tex]\(x^2\)[/tex] and [tex]\(y\)[/tex]).
- Result after addition: [tex]\(3x^2y + 3xy\)[/tex]
- This results into a polynomial, not a monomial, because the terms cannot be combined into a single term due to differing exponents.
2. [tex]\(-12x^2y\)[/tex]:
- The term [tex]\(-12x^2y\)[/tex] has the exact same variables [tex]\(x^2\)[/tex] and [tex]\(y\)[/tex] as [tex]\(3x^2y\)[/tex].
- Result after addition: [tex]\(3x^2y - 12x^2y = -9x^2y\)[/tex]
- This is a single term (monomial).
3. [tex]\(2x^2y^2\)[/tex]:
- The term [tex]\(2x^2y^2\)[/tex] has [tex]\(x^2\)[/tex] and [tex]\(y^2\)[/tex], not matching [tex]\(3x^2y\)[/tex] (which only has [tex]\(y\)[/tex]).
- Result after addition: [tex]\(3x^2y + 2x^2y^2\)[/tex]
- This results into a polynomial, not a monomial, because the terms cannot be combined into a single term due to differing exponents.
4. [tex]\(7xy^2\)[/tex]:
- The term [tex]\(7xy^2\)[/tex] has the variables [tex]\(x\)[/tex] and [tex]\(y^2\)[/tex], not matching [tex]\(3x^2y\)[/tex] (which should have [tex]\(x^2\)[/tex]).
- Result after addition: [tex]\(3x^2y + 7xy^2\)[/tex]
- This results into a polynomial, not a monomial, because the terms cannot be combined into a single term due to differing exponents.
5. [tex]\(-10x^2\)[/tex]:
- The term [tex]\(-10x^2\)[/tex] lacks the variable [tex]\(y\)[/tex] entirely.
- Result after addition: [tex]\(3x^2y - 10x^2\)[/tex]
- This results into a polynomial, not a monomial, because the terms cannot be combined into a single term due to differing variables.
6. [tex]\(4x^2y\)[/tex]:
- The term [tex]\(4x^2y\)[/tex] has the exact same variables [tex]\(x^2\)[/tex] and [tex]\(y\)[/tex] as [tex]\(3x^2y\)[/tex].
- Result after addition: [tex]\(3x^2y + 4x^2y = 7x^2y\)[/tex]
- This is a single term (monomial).
7. [tex]\(3x^3\)[/tex]:
- The term [tex]\(3x^3\)[/tex] has [tex]\(x^3\)[/tex] instead of [tex]\(x^2\)[/tex], not matching [tex]\(3x^2y\)[/tex].
- Result after addition: [tex]\(3x^2y + 3x^3\)[/tex]
- This results into a polynomial, not a monomial, because the terms cannot be combined into a single term due to differing exponents.
### Conclusion:
The terms that would result in a monomial when added to [tex]\(3x^2y\)[/tex] are:
- [tex]\(-12x^2y\)[/tex]
- [tex]\(4x^2y\)[/tex]
So, the correct answers are:
- [tex]\(-12 x^2 y\)[/tex]
- [tex]\(4 x^2 y\)[/tex]
