Problem 39 - Intermediate Algebra

Question

Simplify each rational expression. $$ \frac{2 x+18}{x^{2}-81} $$

Step-by-Step Solution

Answer

Simplified expression is \( \frac{2}{x - 9} \).

Step 1: Identify the Components for Factoring

Look at the rational expression \( \frac{2x+18}{x^2-81} \). Notice that the numerator \( 2x + 18 \) appears to be factorable by taking out the greatest common factor, and the denominator \( x^2 - 81 \) is a difference of squares.

Step 2: Factor the Numerator

Factor the numerator. \( 2x + 18 \) has a greatest common factor of 2. Factoring this out gives: \[ 2x + 18 = 2(x + 9) \]

Step 3: Factor the Denominator

Recognize that \( x^2 - 81 \) is a difference of squares. It can be factored as: \[ x^2 - 81 = (x - 9)(x + 9) \]

Step 4: Write the Expression with the Factored Components

Replace the original numerator and denominator with their factored forms:\[ \frac{2(x + 9)}{(x - 9)(x + 9)} \]

Step 5: Simplify by Cancelling Common Factors

Since \( (x + 9) \) is a common factor in both the numerator and the denominator, it can be cancelled out, leaving:\[ \frac{2}{x - 9} \]Note that \( x eq 9 \) to avoid division by zero.

Key Concepts

Factoring

Factoring is a mathematical technique used to express a number or expression as a product of its factors. It simplifies complex expressions, enabling easier manipulation and solving of equations. In the context of rational expressions, factoring both the numerator and the denominator helps to reduce the expression to its simplest form.

To factor an expression, start by identifying any common factors. In the expression \(2x + 18\), for instance, both terms share a factor of 2. You extract this factor to get the factored form: \(2(x + 9)\).

Factoring transforms expressions, making them simpler and easier to manage. It's about breaking down expressions to their basic building blocks. These blocks can then be used to further simplify or manipulate the overall problem.

Greatest Common Factor

The greatest common factor (GCF) is a key concept in factoring. It represents the highest number that is a factor of two or more numbers or terms. In algebra, finding the GCF often involves breaking down terms into their prime components.

Consider the expression \(2x + 18\). To find its GCF, note the numbers and variables:

• The coefficients are 2 and 18
• The GCF of these coefficients is 2

This means you pull out 2 as the GCF, leaving you with the simplified form: \(2(x + 9)\).

The GCF is vital in ensuring that expressions are factored fully, making it easier to handle algebraic operations like simplifying rational expressions.

Difference of Squares

The difference of squares is a special factoring technique used solely for polynomial expressions in a specific form, \(a^2 - b^2\). This form is notable because it represents a subtraction (difference) of two perfect squares.

For example, \(x^2 - 81\) is a difference of squares since \(81 = 9^2\). The concept simplifies to:

• Identify each square term: \(x^2 = (x)^2\) and \(81 = (9)^2\)
• Apply the formula \(a^2 - b^2 = (a + b)(a - b)\)

Thus, \(x^2 - 81\) becomes \((x + 9)(x - 9)\). This crucial factoring step reveals the expression's underlying structure and is vital in simplifying rational expressions, where similar terms can be canceled out.