Rational Expressions: Simplifying and Finding the Domain

A Rational Expression is simply a fraction where the numerator and denominator are polynomials. Just like numerical fractions, they can be simplified, multiplied, and divided.

Key Rules

1. Finding the Domain (Excluded Values): Since division by zero is undefined, the denominator cannot be zero. Any \(x\)-value that makes the denominator zero is an Excluded Value.

2. Simplifying: Factor both the numerator and denominator completely, then cancel out common factors.

$$\frac{ac}{bc} = \frac{a}{b}, \quad c \neq 0$$

Step-by-Step Example

Simplify the expression and state the domain:

$$\frac{x^2 - 9}{x^2 + 5x + 6}$$

Step 1: Factor the numerator \(x^2 - 9\) is a difference of squares.

$$x^2 - 9 = (x + 3)(x - 3)$$

Step 2: Factor the denominator \(x^2 + 5x + 6\) is a trinomial. We need numbers that multiply to 6 and add to 5.

$$x^2 + 5x + 6 = (x + 2)(x + 3)$$

Step 3: State the Excluded Values (Domain) Set the denominator to zero before canceling.

$$(x + 2)(x + 3) = 0$$

\(x \neq -2\) and \(x \neq -3\). Domain: All real numbers except \(x = -2\) and \(x = -3\).

Step 4: Cancel common factors

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

Cancel \((x + 3)\) from top and bottom.

Solution:

$$\frac{x - 3}{x + 2}, \quad x \neq -2, -3$$

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Why reading isn’t enough

Rational expressions combine factoring with fraction rules, making them a common stumbling block. You must remember to check the domain before simplifying.

Weekzen offers a vast library of rational expression problems. Practice finding excluded values and simplifying complex fractions to build confidence for your next algebra exam.