Introduction to Rational Expressions

Overview

Rational expressions are expressions that involve fractions where the numerator and/or the denominator are polynomials. In this section, we will explore what rational expressions are, how to identify their numerators and denominators, and how to simplify basic rational expressions.

What are Rational Expressions?

A rational expression is any expression that can be written in the form:

\( \frac{P(x)}{Q(x)} \)

Where:

  • P(x): The numerator is a polynomial.
  • Q(x): The denominator is also a polynomial, but it cannot be zero (because division by zero is undefined).

Identifying Numerators and Denominators

In a rational expression, the numerator is the polynomial in the numerator of the fraction, and the denominator is the polynomial in the denominator.

For example, in the rational expression:

\( \frac{x^2 + 3x + 2}{x^2 - 4} \)

  • Numerator: \( x^2 + 3x + 2 \)
  • Denominator: \( x^2 - 4 \)

Simplifying Rational Expressions

To simplify a rational expression, factor both the numerator and the denominator, if possible, and cancel out any common factors.

For example, to simplify:

\( \frac{x^2 - 9}{x^2 - 6x + 9} \)

First, factor the numerator and denominator:

\( \frac{(x - 3)(x + 3)}{(x - 3)(x - 3)} \)

Then, cancel out the common factor of \( (x - 3) \) to get:

\( \frac{x + 3}{x - 3} \)

Practice Questions

  1. Simplify the following rational expression: \( \frac{x^2 - 4}{x^2 - 2x - 8} \)
    Solution

    First, factor both the numerator and the denominator:

    \( \frac{(x - 2)(x + 2)}{(x - 4)(x + 2)} \)

    Then, cancel out the common factor \( (x + 2) \), resulting in:

    \( \frac{x - 2}{x - 4} \)

  2. Identify the numerator and denominator in the rational expression: \( \frac{3x^2 + 5x - 2}{x^2 - 4x + 3} \)
    Solution

    The numerator is:

    \( 3x^2 + 5x - 2 \)

    The denominator is:

    \( x^2 - 4x + 3 \)

  3. Simplify the following rational expression: \( \frac{x^2 - 16}{x^2 - 4x} \)
    Solution

    Factor the numerator and denominator:

    \( \frac{(x - 4)(x + 4)}{x(x - 4)} \)

    Cancel out the common factor \( (x - 4) \), resulting in:

    \( \frac{x + 4}{x} \)

  4. Simplify the following rational expression: \( \frac{x^2 - 25}{x^2 + 5x + 6} \)
    Solution

    Factor both the numerator and the denominator:

    \( \frac{(x - 5)(x + 5)}{(x + 2)(x + 3)} \)

    Since there are no common factors to cancel out, the simplified expression is:

    \( \frac{(x - 5)(x + 5)}{(x + 2)(x + 3)} \)

→Simplifying Rational Expressions