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.

Step 1: Understand Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. To be considered a polynomial, terms can only include variables raised to whole-number powers and coefficients that are real numbers.

Step 2: Analyze Expression (a)

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

Step 3: Analyze Expression (b)

Given expression is \( \frac{\sqrt{x+1}}{2x + 3} \). The numerator \( \sqrt{x+1} \) is not a polynomial because it includes a square root, which is not a whole-number power. Thus, \( \frac{\sqrt{x+1}}{2x + 3} \) is not a rational expression.

Step 4: Analyze Expression (c)

Given expression is \( \frac{x(x^2 - 1)}{x + 3} \). The numerator \( x(x^2 - 1) = x^3 - x \) is a polynomial, and the denominator \( x + 3 \) is a polynomial. Hence, \( \frac{x(x^2 - 1)}{x + 3} \) is a rational expression.

Key Concepts

Polynomials

Polynomials are fundamental building blocks in algebra and are crucial for understanding rational expressions. Simply put, polynomials are expressions made up of variables and coefficients. These variables are raised to whole-number powers (like 0, 1, 2, 3, etc.), and the coefficients are real numbers. A polynomial can have one or multiple terms, each term being a product of a numeric coefficient and a variable raised to a power.

• For example, in the polynomial \(2x^3 - x^2 + 5x - 7\), there are four terms: \(2x^3\), \(-x^2\), \(5x\), and \(-7\).
• In this context, \(2\), \(-1\), \(5\), and \(-7\) are coefficients, and \(x\) is the variable.

Polynomials are easy to work with because they follow predictable patterns and rules, making them easier to manipulate than other algebraic forms. Remember, for an expression to qualify as a polynomial, no variables will appear in denominators, under radical signs (like square roots), or with negative or non-integer exponents.

Numerator and Denominator

In any fraction, including rational expressions, you will encounter a numerator and a denominator. The numerator is the top part of the fraction, and the denominator is the bottom part. For a valid rational expression:

• The numerator must be a polynomial.
• The denominator must also be a polynomial, and importantly, it cannot equal zero.

For instance, consider the expression \(\frac{3x}{x^2 - 1}\):

• The numerator \(3x\) is indeed a polynomial since it includes a variable \(x\) raised to the power of 1.
• The denominator \(x^2 - 1\) is a polynomial as well, as it's a difference of squares, making the entire expression well-defined.

Remember, ensuring that both parts are polynomials is key in verifying rational expressions. Additionally, when evaluating rational expressions, it’s crucial to identify values that might make the denominator zero, as these are undefined in mathematics.

Rational Equations

Rational equations involve rational expressions and come in the form of fractions set equal to each other or a constant. Solving these equations requires a solid understanding of algebraic principles, notably involving the use of the numerator and the denominator.In these equations:- Each part of the rational expression must be a polynomial.- To solve, you often need to find a common denominator to simplify or eliminate the fractions. For example, let's solve the rational equation \(\frac{x + 1}{2} = \frac{x - 3}{4}\).1. Find a common denominator (here, it's 4) and multiply each term by it to clear the fractions: \[ 4 \cdot \frac{x + 1}{2} = 4 \cdot \frac{x - 3}{4} \] This simplifies to: \[ 2(x + 1) = x - 3 \]2. Solve the resulting linear equation. Distribute and then combine like terms to isolate the variable.By learning and practicing with rational equations, students gain the ability to manipulate more complex algebraic topics fluently. Remember to check that the solution doesn't make the original denominator zero!