Problem 1 - Precalculus: Mathematics for Calculus

Question

Which of the following are rational expressions? (a) \(\frac{3 x}{x^{2}-1}\) (b) \(\frac{\sqrt{x+1}}{2 x+3}\) (c) \(\frac{x\left(x^{2}-1\right)}{x+3}\)

Step-by-Step Solution

Answer

Expressions (a) and (c) are rational expressions.

Step 1: Understanding Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. To determine if a given expression is rational, we need to verify that both the top and bottom of the fraction are polynomials.

Step 2: Analyzing Expression (a)

Expression (a) is \( \frac{3x}{x^2-1} \). In this expression, the numerator is a polynomial "3x", and the denominator \( x^2 - 1 \) is also a polynomial. Therefore, expression (a) is a rational expression.

Step 3: Analyzing Expression (b)

Expression (b) is \( \frac{\sqrt{x+1}}{2x+3} \). Here, while the denominator \( 2x + 3 \) is a polynomial, the numerator \( \sqrt{x+1} \) is not a polynomial because it contains a square root function. Therefore, expression (b) is not a rational expression.

Step 4: Analyzing Expression (c)

Expression (c) is \( \frac{x(x^2-1)}{x+3} \). The numerator is \( x(x^2-1) = x^3 - x \), which is a polynomial. The denominator \( x + 3 \) is also a polynomial. Thus, expression (c) is a rational expression.

Key Concepts

Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients, structured using addition, subtraction, and multiplication. Each term in a polynomial features a variable raised to a whole number exponent. For example, in the expression \( 3x^2 + 4x - 1 \), each part, like \( 3x^2 \), is a term. Here are key aspects of polynomials:

• Polynomials can be of various degrees, with the degree being the highest exponent in the expression.
• They are composed of terms such as constants (e.g., \( -1 \)) or multiples of variables (e.g., \( 4x \)).
• The structure involves operations of addition, subtraction, and multiplication.
• Division by variables is not part of polynomial operations.

It's crucial to focus on the fact that only whole number exponents are allowed. This distinguishes polynomials from other expressions like those that include roots or fractional exponents.

Numerator and Denominator

In any fraction or expression written in fractional form, the numerator and denominator play critical roles. The numerator is the top part, while the denominator is the bottom part of the fraction. When it comes to rational expressions, understanding these parts is vital:

• The numerator: It indicates how many parts out of the whole are being considered, represented by a polynomial in rational expressions.
• The denominator: This dictates the total number of equal parts that make up the whole. It must not be zero; otherwise, the expression becomes undefined.

Recognizing that both the numerator and denominator must be polynomials helps in identifying rational expressions. For instance, in \( \frac{3x}{x^2-1} \), both parts are polynomials, making it a correct rational expression.

Square Root Functions

Square root functions involve the operation of taking the square root of a number or an algebraic expression. Unlike polynomials, square roots transform exponents into fractions, which fundamentally affects their classification:

• Square roots are not polynomials because they introduce non-whole number exponents; for example, \( \sqrt{x} \) is equivalent to \( x^{0.5} \).
• This makes simplifying expressions more complex, as they often require careful handling to rationalize denominators.
• In the context of rational expressions, if a square root is in the numerator or denominator, this alters its status from a polynomial to a non-polynomial part.

Thus, expressions involving square roots in either the numerator or the denominator, like \( \frac{\sqrt{x+1}}{2x+3} \), cannot be considered rational expressions due to the presence of the non-polynomial element introduced by the square root.