Page 2: Operations on Rational Expressions

This page focuses on operations involving rational expressions, including addition, subtraction, multiplication, and division.

Vocabulary: Rational expressions are fractions where both the numerator and denominator are polynomials.

The page provides detailed examples of how to perform these operations, emphasizing the importance of finding common denominators and simplifying results.

Example: When adding fractions like $x + 1$ / $3 - 2x$ and $x - 2$ / $x² - 4x$, students must find a common denominator before adding.

The page also covers the process of simplifying rational expressions by factoring and canceling common terms.

Highlight: When performing operations on rational expressions, always check for restrictions on the domain to ensure the denominator is never zero.

Page 3: Advanced Problems and Techniques

This page presents more complex problems involving rational functions and expressions, demonstrating advanced techniques for simplification and problem-solving.

Example: One problem involves simplifying the complex fraction $x² + x + 3$ / $4x² + 11x - 3$ - $(2x² + 1) - (x + 3)$ / $(4x - 1)(4x + 1)$.

The page also covers polynomial long division, which is useful when simplifying certain types of rational expressions.

Highlight: When dealing with complex fractions, it's often helpful to separate the expression into simpler parts and simplify each part individually before combining them.

The final section of the page includes problems that require a combination of techniques, such as factoring, finding common denominators, and simplifying.

Vocabulary: The domain of a rational function includes all real numbers except those that make the denominator equal to zero.

Throughout the page, emphasis is placed on carefully checking the domain restrictions for each step of the simplification process.

Page 1: Introduction to Rational Functions

This page introduces the concept of rational functions and provides examples of finding their domains.

Definition: A rational function is a function of the form F$x$ = N$x$ / G$x$, where N$x$ and G$x$ are polynomials and G$x$ ≠ 0.

The page presents several exercises on determining the domain of rational functions. It emphasizes the importance of finding values that make the denominator equal to zero, as these are excluded from the domain.

Example: For the function F$x$ = $x - 9$ / $x³ + 8x² - x + 8$, students must solve the equation x³ + 8x² - x + 8 = 0 to find the excluded values.

The page also covers the evaluation of rational functions at specific points, such as F$√15 + 2$.

Highlight: When solving for domains, it's crucial to factor polynomials and identify all possible zero values in the denominator.