Chapter 6, Problem 1TYWP

To determine

The correct statement that defines a rational expression from the following choices,

A. An algebraic expression made up of a term or the sum of a finite number of terms with real coefficients and whole number exponents.

B. A polynomial equation of degree two.

C. An expression with one or more fractions in the numerator or denominator or both

D. The quotient of two polynomials with denominator not zero.

Answer to Problem 1TYWP

Solution:

The correct option is D. The quotient of two polynomials with denominator not zero.

Explanation of Solution

Given Information:

The choices are:

A. An algebraic expression made up of a term or the sum of a finite number of terms with real coefficients and whole number exponents.

B. A polynomial equation of degree two.

C. An expression with one or more fractions in the numerator or denominator or both

D. The quotient of two polynomials with denominator not zero.

A rational expression is an expression of the form $\frac{P}{Q}$ where $P,Q$ are polynomials and $Q\ne 0$.

According to option A, a rational expression is an algebraic expression made up of a term or the sum of a finite number of terms with real coefficients and whole number exponents.

Therefore, the expression $\frac{x^2+5x}{0}$ is also a rational number as the numerator is the sum of a finite number of terms with real coefficients and whole number exponents and the denominator is a constant term.

But by the definition of the rational expression, the denominator cannot be zero.

Hence, option A is incorrect.

According to option B, a rational expression is a polynomial equation of degree two.

As the polynomial with degree greater than 2 or less than two are also rational expressions. Thus, the statement of option B is not the correct definition of rational expression.

According to option C, a rational expression is an expression with one or more fractions in the numerator or denominator or both.

Therefore, the expression $\frac{\left(\frac{x}{3}\right)}{\left(\frac{0}{5}\right)}$ is also a rational number as the numerator and the denominator both are fractions.

But by the definition of the rational expression, the denominator cannot be zero.

Hence, option C is incorrect.

According to option D, a rational expression is the quotient of two polynomials with denominator not zero.

For example, $\frac{-6x}{x^3+8}$ is a rational expression.

As the statement satisfies the definition of rational expression. Hence, the option D is correct definition.