Simplifying Rational Expressions

Overview

A rational expression is a fraction in which the numerator and/or denominator is a polynomial. Simplifying a rational expression involves reducing it to its simplest form. This can be done by factoring both the numerator and denominator, canceling out common factors, and then simplifying the result.

Steps to Simplify Rational Expressions

To simplify a rational expression, follow these steps:

  1. Factor the numerator and denominator.
  2. Cancel out any common factors between the numerator and denominator.
  3. Simplify the remaining expression.

Example 1: Simplifying a Rational Expression

Consider the rational expression:

\( \frac{2x^2 + 6x}{4x} \)

Step 1: Factor the numerator and denominator:

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

Step 2: Cancel out the common factor of \( x \):

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

Step 3: Simplify the remaining expression:

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

So, the simplified rational expression is:

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

Example 2: Simplifying a More Complex Rational Expression

Consider the expression:

\( \frac{x^2 - 9}{x^2 - 4x} \)

Step 1: Factor both the numerator and denominator:

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

Step 2: Cancel out the common factor of \( (x - 4) \):

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

So, the simplified rational expression is:

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

Practice Questions

  1. Question 1: Simplify the following rational expression:

    \( \frac{x^2 - 16}{x^2 - 4x} \)

    Solution

    Step 1: Factor both the numerator and denominator:

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

    Step 2: Cancel out the common factor of \( (x - 4) \):

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

    So, the simplified rational expression is:

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

  2. Question 2: Simplify the following rational expression:

    \( \frac{6x^2 - 18x}{12x^2} \)

    Solution

    Step 1: Factor both the numerator and denominator:

    \( \frac{6x(x - 3)}{12x^2} \)

    Step 2: Cancel out the common factor of \( 6x \):

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

    So, the simplified rational expression is:

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

  3. Question 3: Simplify the following rational expression:

    \( \frac{x^2 + 5x + 6}{x^2 + 3x} \)

    Solution

    Step 1: Factor both the numerator and denominator:

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

    Step 2: Cancel out the common factor of \( (x + 3) \):

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

    So, the simplified rational expression is:

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

  4. Question 4: Simplify the following rational expression:

    \( \frac{3x^2 + 6x}{9x} \)

    Solution

    Step 1: Factor both the numerator and denominator:

    \( \frac{3x(x + 2)}{9x} \)

    Step 2: Cancel out the common factor of \( 3x \):

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

    So, the simplified rational expression is:

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

→Multiplying and Dividing Rational Expressions