Problem 27 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \text { perform the indicated operations and simplify. } $$ $$ \frac{12}{x^{2}+2 x}+\frac{4}{x}+\frac{2}{x+2} $$

Step-by-Step Solution

Answer

\( \frac{2(3x + 10)}{x(x+2)} \) is the simplified form.

Step 1: Identify the Least Common Denominator (LCD)

First, identify the Least Common Denominator of the fractions. The denominators are \( x^2 + 2x \), \( x \), and \( x + 2 \). Factoring \( x^2 + 2x \) gives \( x(x+2) \). Thus, the LCD is \( x(x+2) \).

Step 2: Rewrite Each Fraction with the LCD

Rewrite each fraction with the LCD as the denominator. For \( \frac{12}{x^2 + 2x} \):\[\frac{12}{x(x+2)} = \frac{12}{x(x+2)}\] (already has the LCD).For \( \frac{4}{x} \):\[\frac{4}{x} = \frac{4(x+2)}{x(x+2)}\]For \( \frac{2}{x+2} \):\[\frac{2}{x+2} = \frac{2x}{x(x+2)}\]

Step 3: Combine the Fractions

Combine the fractions over the common denominator:\[\frac{12}{x(x+2)} + \frac{4(x+2)}{x(x+2)} + \frac{2x}{x(x+2)} = \frac{12 + 4(x+2) + 2x}{x(x+2)}\]

Step 4: Simplify the Numerator

Expand and simplify the numerator:\[12 + 4(x+2) + 2x = 12 + 4x + 8 + 2x = 6x + 20\]Thus, we have:\[\frac{6x + 20}{x(x+2)}\]

Step 5: Simplify the Fraction

Factor the numerator if possible to simplify the expression:\[6x + 20 = 2(3x + 10)\]Now the fraction is:\[\frac{2(3x + 10)}{x(x+2)}\]Since there are no common factors in the numerator and denominator, the expression is simplified.

Key Concepts

Understanding the Least Common Denominator

When working with rational expressions, one critical step is finding the Least Common Denominator (LCD). The LCD helps combine fractions that have different denominators.
The LCD is essentially the smallest multiple that all denominators can divide into without leaving a remainder.
In the exercise, the denominators involved are polynomials like:

• \( x^2 + 2x \)
• \( x \)
• \( x + 2 \)

To find the LCD, begin by factoring each denominator.
The polynomial \( x^2 + 2x \) can be factored into \( x(x+2) \). Thus, both \( x \) and \( x + 2 \) are factors of this polynomial.
This means the LCD is \( x(x+2) \) as it includes all necessary factors needed for each fraction's denominator.

The Essentials of Factoring Polynomials

Factoring polynomials is a process where a polynomial is expressed as the product of smaller polynomials.
It can simplify calculations greatly, especially when dealing with rational expressions. The goal in factoring is to identify common factors that can be multiplied together to produce the original polynomial.
In our exercise, we have the polynomial \( x^2 + 2x \). Factoring begins by looking for common terms.

• In this case, both terms \( x^2 \) and \( 2x \) have \( x \) as a common factor.
  This leads us to rearrange the polynomial as \( x(x + 2) \).
• With this factorization, calculations become straightforward, as fewer computations are needed when simplifying or finding the LCD.
  Factoring is an indispensable skill in algebra that assists in navigating complex expressions.

Mastering Fraction Operations with Rational Expressions

Working with fractions that include variables requires understanding operations like addition, subtraction, and simplification.
Once you have a common denominator, you can easily add or subtract rational expressions.

• First, ensure all fractions are rewritten to have the common denominator.
  For example, in our exercise:
• \( \frac{12}{x^2 + 2x} = \frac{12}{x(x+2)} \) (already in terms of the LCD)
• \( \frac{4}{x} = \frac{4(x+2)}{x(x+2)} \)
• \( \frac{2}{x+2} = \frac{2x}{x(x+2)} \)
• Next, combine these fractions into a single expression:
  \( \frac{12 + 4(x+2) + 2x}{x(x+2)} \)
• Simplifying involves combining like terms:
  \( 12 + 4x + 8 + 2x = 6x + 20 \)
• Finally, check for any further simplification by factoring:
  \( 6x + 20 = 2(3x + 10) \)
  The fraction is simplified as \( \frac{2(3x+10)}{x(x+2)} \), assuming no common factors remain.

Fraction operations with rational expressions become manageable when you break them down step-by-step.