Study Notes on Rational Expressions

Definition of Rational Expression

• A rational expression is defined as a fraction where both the numerator and denominator are polynomials.

General Form of Rational Expressions

• The general form of a rational expression can be represented as:
  • $f(x) = \frac{P(x)}{Q(x)}$
  • Where:
  • $f(x)$ represents the entire rational expression.
  • $P(x)$ represents the polynomial in the numerator.
  • $Q(x)$ represents the polynomial in the denominator.

Examples of Rational Expressions

• Example 1: Consider the following rational expression:

  • It contains both a polynomial in the numerator and a polynomial in the denominator.
  • A graph is provided for visualization purposes.
  • The details of the graph are not crucial at this point.

• Example 2: A second rational expression is analyzed:

  • Note that a term without any variables can still qualify as a polynomial, provided it meets the requirements of a polynomial.
  • Both the numerator and the denominator contain polynomials, continuing to validate this as a rational expression.
  • Another graph is provided for visual reference.

Non-Example of Rational Expression

• Example 3: This is not a rational expression:
  • The numerator is a polynomial.
  • However, the denominator includes division by a variable.
  • Since this does not meet the polynomial criteria in the denominator, this expression is classified as not a rational expression.

Summary of Key Points

• A rational expression is defined as a fraction with both numerator and denominator as polynomials.
• Understanding the rules of polynomials is essential for determining whether an expression qualifies as a rational expression or not.
• The visual representations (graphs) help in grasping the concept but the specifics of these graphs are not necessary for comprehension at this level.