Problem 11 - Algebra for College Students

Question

Perform the indicated operations, and express your answers in simplest form. $$ \frac{5}{x}-\frac{5 x-30}{x^{2}+6 x}+\frac{x}{x+6} $$

Step-by-Step Solution

Answer

The simplest form is \( \frac{x^2 + 60}{x(x+6)} \).

Step 1: Factor the Denominator

Identify and factor the denominator of the second fraction, \[ x^2 + 6x = x(x + 6) \]This helps to find a common denominator for all fractions.

Step 2: Find the Least Common Denominator (LCD)

The denominators of the fractions are \( x \), \( x(x+6) \), and \( x+6 \). Therefore, the least common denominator is \[ x(x+6) \].

Step 3: Convert Each Fraction to Use the LCD

Multiply each fraction so that they all have \( x(x+6) \) as the denominator.1. \( \frac{5}{x} = \frac{5(x+6)}{x(x+6)} \)2. Already having \( \frac{5x-30}{x(x+6)} \) 3. \( \frac{x}{x+6} = \frac{x \cdot x}{x(x+6)} = \frac{x^2}{x(x+6)} \)

Step 4: Combine the Fractions

Add or subtract the numerators of the fractions:\[\frac{5(x+6) - (5x-30) + x^2}{x(x+6)}\]Simplify the expression inside the numerator.

Step 5: Simplify the Numerator

Expand and combine terms in the numerator:\[5(x+6) - (5x - 30) + x^2 = 5x + 30 - 5x + 30 + x^2 = x^2 + 60\]

Step 6: Write the Simplified Expression

Now express the final combined fraction as:\[\frac{x^2 + 60}{x(x+6)}\]This fraction is in its simplest form.

Key Concepts

Least Common Denominator

When working with algebraic fractions, finding the Least Common Denominator (LCD) is crucial because it allows you to combine the fractions into a single expression. The LCD is essentially the smallest expression that all the denominators of the fractions can divide into without leaving a remainder.

For example, when dealing with fractions that have denominators like \(x\), \(x(x+6)\), and \(x+6\), the LCD would be \(x(x+6)\). Here's why:

• \(x\) can divide into \(x(x+6)\) because \(x\) is a factor of \(x(x+6)\).
• \(x(x+6)\) is already an exact match.
• \(x+6\) can divide into \(x(x+6)\) because \(x+6\) is a factor of \(x(x+6)\).

Finding the LCD simplifies the process of adding or subtracting fractions because it provides a shared base to work from. Once the LCD has been identified, each fraction can be rewritten with the common denominator, making it easy to perform the arithmetic operations required.

Factoring Polynomials

Factoring polynomials is an essential skill when working with algebraic fractions. By breaking down a polynomial into the product of its simplest factors, we can reveal its structure and make operations like finding common denominators much easier.

Let's consider the polynomial \(x^2 + 6x\). To factor this polynomial, we look for common factors in each term. Here, both terms share \(x\), so the polynomial factors to \(x(x + 6)\).

• \(x^2\) becomes \(x \cdot x\).
• \(6x\) is already in its simplest form as \(x \cdot 6\).

This factoring process is crucial in simplifying your work with fractions, because it reveals the full expression of a denominator, making it easier to determine the least common denominator for multiple fractions.

Simplifying Expressions

Simplifying expressions involves combining like terms and reducing the expression to its simplest form. This step is often one of the last in solving problems involving algebraic fractions.

In the given exercise, after aligning the fractions under a common denominator, the next task is to simplify the numerators. Consider the numerator: \(5(x+6) - (5x-30) + x^2\).

Steps:

• Expand \(5(x+6)\) to get \(5x + 30\).
• Distribute the minus sign: \(-(5x - 30)\) becomes \(-5x + 30\).
• Combine terms: \(5x + 30 - 5x + 30 + x^2\).
• This simplifies to \(x^2 + 60\) as the terms \(5x - 5x\) cancel each other out, leaving \(x^2 + 60\).

The use of these steps allows you to express the problem's answer in a compact form and ensures that the fraction is indeed in its simplest, most manageable form.