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Adding which terms to [tex]$3x^2y$[/tex] would result in a monomial? Check all that apply.

A. [tex]3xy[/tex]
B. [tex]-12x^2y[/tex]
C. [tex]2x^2y^2[/tex]
D. [tex]7xy^2[/tex]
E. [tex]-10x^2[/tex]
F. [tex]4x^2y[/tex]
G. [tex]3x^3[/tex]

Sagot :

GinnyAnsweridn
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]
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