Adding Which Terms To $3x^2y$ Would Result In A Monomial? Check All That Apply.A. $3xy$ B. $-12x^2y$ C. $2x^2y^2$ D. $7xy^2$ E. $-10x^2$ F. $4x^2y$ G. $3x^3$

In algebra, a monomial is a type of polynomial expression that consists of only one term. It is a product of variables and coefficients, where the variables are raised to non-negative integer powers. In this article, we will explore which terms can be added to the expression $3x^2y$ to result in a monomial.

What is a Monomial?

A monomial is a polynomial expression that consists of only one term. It can be written in the form $ax^n$, where $a$ is a coefficient and $n$ is a non-negative integer. For example, $3x^2y$ is a monomial because it consists of only one term, which is the product of the coefficient $3$, the variable $x$ raised to the power of $2$, and the variable $y$.

Adding Terms to $3x^2y$

To determine which terms can be added to $3x^2y$ to result in a monomial, we need to consider the properties of monomials. A monomial must have only one term, and the variables in the term must be raised to non-negative integer powers.

Option A: $3xy$

The term $3xy$ has only one variable, $x$, raised to the power of $1$, and the variable $y$ is also raised to the power of $1$. This term can be added to $3x^2y$ to result in a monomial.

Option B: $-12x^2y$

The term $-12x^2y$ has the same variables as $3x^2y$, but with a different coefficient. This term can be added to $3x^2y$ to result in a monomial.

Option C: $2x^2y^2$

The term $2x^2y^2$ has the same variables as $3x^2y$, but with a different exponent for the variable $y$. This term cannot be added to $3x^2y$ to result in a monomial because it has two variables raised to non-negative integer powers.

Option D: $7xy^2$

The term $7xy^2$ has the variable $x$ raised to the power of $1$, but the variable $y$ is raised to the power of $2$. This term cannot be added to $3x^2y$ to result in a monomial because it has two variables raised to non-negative integer powers.

Option E: $-10x^2$

The term $-10x^2$ has the same variable as $3x^2y$, but with a different coefficient and a different exponent for the variable $x$. This term cannot be added to $3x^2y$ to result in a monomial because it has a different exponent for the variable $x$.

Option F: $4x^2y$

The term $4x^2y$ has the same variables as $3x^2y$, but with a different coefficient. This term can be added to $3x^2y$ to result in a monomial.

Option G: $3x^3$

The term $3x^3$ has the same variable as $3x^2y$, but with a different exponent for the variable $x$. This term cannot be added to $3x^2y$ to result in a monomial because it has a different exponent for the variable $x$.

Conclusion

In conclusion, the terms that can be added to $3x^2y$ to result in a monomial are:

• Option A: $3xy$
• Option B: $-12x^2y$
• Option F: $4x^2y$

These terms can be added to $3x^2y$ to result in a monomial because they have the same variables and exponents, and the variables are raised to non-negative integer powers. The other options cannot be added to $3x^2y$ to result in a monomial because they have different exponents for the variables or two variables raised to non-negative integer powers.

Final Answer

The final answer is:

• Option A: $3xy$
• Option B: $-12x^2y$
• Option F: $4x^2y$

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In our previous article, we explored which terms can be added to the expression $3x^2y$ to result in a monomial. In this article, we will answer some frequently asked questions about monomials and polynomial terms.

Q: What is the difference between a monomial and a polynomial?

A: A monomial is a type of polynomial expression that consists of only one term. A polynomial, on the other hand, is a mathematical expression that consists of two or more terms. For example, $3x^2y$ is a monomial, while $3x^2y + 4x^2y$ is a polynomial.

Q: What are the properties of a monomial?

A: A monomial must have only one term, and the variables in the term must be raised to non-negative integer powers. For example, $3x^2y$ is a monomial because it consists of only one term, and the variables $x$ and $y$ are raised to non-negative integer powers.

Q: Can a monomial have a variable raised to a negative power?

A: No, a monomial cannot have a variable raised to a negative power. The variables in a monomial must be raised to non-negative integer powers. For example, $3x^{-2}y$ is not a monomial because the variable $x$ is raised to a negative power.

Q: Can a monomial have a variable raised to a fractional power?

A: No, a monomial cannot have a variable raised to a fractional power. The variables in a monomial must be raised to non-negative integer powers. For example, $3x^{1/2}y$ is not a monomial because the variable $x$ is raised to a fractional power.

Q: Can a monomial have a variable raised to a power of zero?

A: Yes, a monomial can have a variable raised to a power of zero. For example, $3x^0y$ is a monomial because the variable $x$ is raised to a power of zero.

Q: Can a monomial have a coefficient of zero?

A: Yes, a monomial can have a coefficient of zero. For example, $0x^2y$ is a monomial because it consists of only one term, and the coefficient is zero.

Q: Can a monomial have a variable with no exponent?

A: Yes, a monomial can have a variable with no exponent. For example, $3xy$ is a monomial because it consists of only one term, and the variable $y$ has no exponent.

Q: Can a monomial have a variable with a negative exponent?

A: No, a monomial cannot have a variable with a negative exponent. The variables in a monomial must be raised to non-negative integer powers.

Q: Can a monomial have a variable with a fractional exponent?

A: No, a monomial cannot have a variable with a fractional exponent. The variables in a monomial must be raised to non-negative integer powers.

Conclusion

In conclusion, a monomial is a type of polynomial expression that consists of only one term, and the variables in the term must be raised to non-negative integer powers. A monomial can have a variable raised to a power of zero, but it cannot have a variable raised to a negative power or a fractional power. We hope this article has helped to clarify the properties of monomials and polynomial terms.

Final Answer

The final answer is:

• A monomial is a type of polynomial expression that consists of only one term.
• The variables in a monomial must be raised to non-negative integer powers.
• A monomial can have a variable raised to a power of zero.
• A monomial cannot have a variable raised to a negative power or a fractional power.