The Expression in Question

Oct 13, 2025 by wbrum.org 24 views

What is a Polynomial?

A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. It can be written in the form of a sum of terms, where each term is a product of a coefficient and one or more variables raised to non-negative integer powers.

The Expression in Question

The given expression is $\frac{5}{2y^2} + 7y$ . To determine whether this expression is a polynomial, we need to analyze its components.

Breaking Down the Expression

The expression consists of two terms:

$\frac{5}{2y^2}$
$7y$

Analyzing the First Term

The first term, $\frac{5}{2y^2}$ , can be rewritten as $\frac{5}{2} \cdot \frac{1}{y^2}$ . This term has a coefficient of $\frac{5}{2}$ and a variable $y$ raised to the power of $-2$ . Since the power is negative, this term is not a polynomial.

Analyzing the Second Term

The second term, $7y$ , has a coefficient of $7$ and a variable $y$ raised to the power of $1$ . This term meets the criteria of a polynomial.

Conclusion

Since the expression contains a term with a negative power, it is not a polynomial. Therefore, the correct answer is:

A. Not a polynomial

Why is the Expression Not a Polynomial?

The expression is not a polynomial because it contains a term with a negative power, which is not allowed in a polynomial expression. A polynomial can only have non-negative integer powers of variables.

What are the Characteristics of a Polynomial?

A polynomial must meet the following criteria:

It must be a sum of terms.
Each term must be a product of a coefficient and one or more variables raised to non-negative integer powers.
It cannot contain any negative powers of variables.

Examples of Polynomials

Here are some examples of polynomials:

$2x + 3y$
$x^2 + 4y^2$
$3x^2y - 2xy^2$

Examples of Non-Polynomials

Here are some examples of non-polynomials:

$\frac{1}{x}$
$x^{-2}$
$\frac{1}{x^2}$

Conclusion

Q: What is a polynomial?

A: A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. It can be written in the form of a sum of terms, where each term is a product of a coefficient and one or more variables raised to non-negative integer powers.

Q: What are the characteristics of a polynomial?

A: A polynomial must meet the following criteria:

It must be a sum of terms.
Each term must be a product of a coefficient and one or more variables raised to non-negative integer powers.
It cannot contain any negative powers of variables.

Q: What are some examples of polynomials?

A: Here are some examples of polynomials:

$2x + 3y$
$x^2 + 4y^2$
$3x^2y - 2xy^2$

Q: What are some examples of non-polynomials?

A: Here are some examples of non-polynomials:

$\frac{1}{x}$
$x^{-2}$
$\frac{1}{x^2}$

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

A: No, a polynomial cannot have a variable raised to a negative power. The power of a variable in a polynomial must be a non-negative integer.

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

A: No, a polynomial cannot have a variable raised to a fractional power. The power of a variable in a polynomial must be a non-negative integer.

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

A: Yes, a polynomial can have a variable raised to a power of zero. For example, the polynomial $x^2 + 4y^2$ has a term $4y^2$ , where $y$ is raised to the power of 2, and the polynomial $2x + 3y$ has a term $3y$ , where $y$ is raised to the power of 1, and the polynomial $4$ has a term $4$ , where the variable is raised to the power of 0.

Q: Can a polynomial have a constant term?

A: Yes, a polynomial can have a constant term. For example, the polynomial $2x + 3y + 4$ has a constant term of 4.

Q: Can a polynomial have a coefficient of zero?

A: Yes, a polynomial can have a coefficient of zero. For example, the polynomial $2x + 0y$ has a coefficient of zero for the term $0y$ .

Q: Can a polynomial have a variable with no coefficient?

A: Yes, a polynomial can have a variable with no coefficient. For example, the polynomial $x + 3y$ has a variable $x$ with no coefficient.

Q: Can a polynomial have multiple variables?

A: Yes, a polynomial can have multiple variables. For example, the polynomial $2x + 3y + 4z$ has three variables: $x$ , $y$ , and $z$ .

Q: Can a polynomial have a variable raised to a power that is a product of two or more variables?

A: Yes, a polynomial can have a variable raised to a power that is a product of two or more variables. For example, the polynomial $x^2y^2 + 3x^2y$ has a term $x^2y^2$ , where the variable $x$ is raised to the power of 2 and the variable $y$ is raised to the power of 2.

Q: Can a polynomial have a variable raised to a power that is a product of two or more variables, where one of the variables is raised to a negative power?

A: No, a polynomial cannot have a variable raised to a power that is a product of two or more variables, where one of the variables is raised to a negative power. The power of a variable in a polynomial must be a non-negative integer.

Q: Can a polynomial have a variable raised to a power that is a product of two or more variables, where one of the variables is raised to a fractional power?

A: No, a polynomial cannot have a variable raised to a power that is a product of two or more variables, where one of the variables is raised to a fractional power. The power of a variable in a polynomial must be a non-negative integer.