Question

Which of the following are rational expressions?

(a)$\frac{3x}{x^2-1}$

(b)$\frac{\sqrt{x+1}}{2x+3}$

(c)$\frac{x\left(x^2-1\right)}{x+3}$

Short answer

(a)$\frac{3x}{x^2-1}$is rationalexpression

(b)$\frac{\sqrt{x+1}}{2x+3}$is not rational expression.

(c)$\frac{x\left(x^2-1\right)}{x+3}$is rationalexpression.

Step-by-step solution

Part a. Step 1. Apply the concept of rational expressions.

A rational expression is a fractional expression in which both the numerator and the denominator are polynomials.

Part a. Step 2. Analyze the expression

Given $\frac{3x}{x^2-1}$

Here the numerator is $3x$and the denominator is $x^2-1$.

Part a. Step 3. Find whether rational expression or not.

Since the numerator and denominator are both polynomials,$\frac{3x}{x^2-1}$ is rational expression.

Part b. Step 1. Apply the concept of rational expressions.

A rational expression is a fractional expression in which both the numerator and the denominator are polynomials.

Part b. Step 2. Analyze the expression.

Given $\frac{\sqrt{x+1}}{2x+3}$

Here the numerator is$\sqrt{x+1}$ and the denominator is $2x+3$.

Part b. Step 3. Find whether rational expression or not.

Since the numerator$\sqrt{x+1}$ is a radical and not a polynomial $\frac{\sqrt{x+1}}{2x+3}$, is not rational expression.

Part c. Step 1. Apply the concept of rational expressions.

A rational expression is a fractional expression in which both the numerator and the denominator are polynomials.

Part c. Step 2. Analyze the expression.

Given $\frac{x\left(x^2-1\right)}{x+3}$

Here the numerator is$x\left(x^2-1\right)$ and the denominator is $x+3$.

Part c. Step 3. Find whether rational expression or not.

Since the numerator and denominator are both polynomials,$\frac{x\left(x^2-1\right)}{x+3}$ is rational expression.