Key Concepts of Rational Expressions to Know for Algebra and Trigonometry

Rational expressions are fractions with polynomials in the numerator and denominator, where the denominator can't be zero. Understanding how to simplify, multiply, divide, and solve these expressions is key to mastering algebra and trigonometry concepts.

1. Definition of rational expressions

   • A rational expression is a fraction where the numerator and denominator are both polynomials.
   • The denominator cannot be zero, as this would make the expression undefined.
   • Examples include expressions like (x^2 + 3)/(x - 1).

2. Simplifying rational expressions

   • To simplify, factor both the numerator and denominator completely.
   • Cancel out any common factors between the numerator and denominator.
   • Always check for restrictions on the variable that could make the denominator zero.

3. Multiplying rational expressions

   • Multiply the numerators together and the denominators together.
   • Simplify the resulting expression by canceling any common factors.
   • Remember to check for restrictions on the variable in the original expressions.

4. Dividing rational expressions

   • To divide, multiply by the reciprocal of the second expression.
   • Follow the same steps as multiplication: multiply numerators and denominators, then simplify.
   • Ensure to check for restrictions on the variable.

5. Adding and subtracting rational expressions

   • Find a common denominator before adding or subtracting.
   • Rewrite each expression with the common denominator, then combine the numerators.
   • Simplify the resulting expression and check for restrictions.

6. Finding the least common denominator (LCD)

   • The LCD is the smallest expression that can be used as a common denominator for all rational expressions involved.
   • Factor each denominator to identify the highest powers of all prime factors.
   • Use the LCD to rewrite each rational expression for addition or subtraction.

7. Solving rational equations

   • Clear the fractions by multiplying both sides by the LCD.
   • Solve the resulting polynomial equation for the variable.
   • Check for extraneous solutions that may arise from the original denominators.

8. Domain of rational expressions

   • The domain consists of all real numbers except those that make the denominator zero.
   • Identify restrictions by setting the denominator equal to zero and solving for the variable.
   • Express the domain in interval notation, excluding the restricted values.

9. Graphing rational functions

   • Identify key features such as intercepts, asymptotes, and holes.
   • Plot points by substituting values into the rational function.
   • Use the behavior of the function near asymptotes to sketch the graph accurately.

10. Asymptotes of rational functions

    • Vertical asymptotes occur where the denominator is zero (and the numerator is not).
    • Horizontal asymptotes are determined by the degrees of the numerator and denominator.
    • Oblique (slant) asymptotes may exist if the degree of the numerator is one higher than that of the denominator.