Problem 34 - Elementary Algebra

Question

For Problems 1-40, perform the indicated operations and express answers in simplest form. $$ \frac{1}{x^{2}+7 x}-\frac{2}{x^{2}-7 x}-\frac{5}{x^{2}-49} $$

Step-by-Step Solution

Answer

The simplified form is \(\frac{-3(2x + 7)}{x(x+7)(x-7)}\).

Step 1: Identify Least Common Denominator (LCD)

First, identify the least common denominator (LCD) of the fractions. The denominators are \(x^2 + 7x\), \(x^2 - 7x\), and \(x^2 - 49\). These can be factored as: \(x(x + 7)\), \(x(x - 7)\), and \((x + 7)(x - 7)\). The LCD is \(x(x + 7)(x - 7)\).

Step 2: Rewrite Fractions with LCD

Convert each fraction to have the same denominator, the LCD from Step 1:1. \(\frac{1}{x^2 + 7x} = \frac{1}{x(x+7)}\) is multiplied by \(x-7\) to get \(\frac{x-7}{x(x+7)(x-7)}\).2. \(\frac{2}{x^2 - 7x} = \frac{2}{x(x-7)}\) is multiplied by \(x+7\) to get \(\frac{2(x+7)}{x(x+7)(x-7)}\).3. \(\frac{5}{x^2 - 49} = \frac{5}{(x+7)(x-7)}\) is multiplied by \(x\) to get \(\frac{5x}{x(x+7)(x-7)}\).

Step 3: Subtract and Simplify

Combine the fractions under a single denominator and simplify:\[\frac{x-7}{x(x+7)(x-7)} - \frac{2(x+7)}{x(x+7)(x-7)} - \frac{5x}{x(x+7)(x-7)}\]Simplify the numerator:\(x-7 - 2(x+7) - 5x = x - 7 - 2x - 14 - 5x = -6x - 21\).Therefore, the simplified expression is:\[\frac{-6x - 21}{x(x+7)(x-7)}\]

Step 4: Factor and Simplify Further

Look for common factors in the numerator. Factor out a \(-3\) from \(-6x - 21\):\[\frac{-3(2x + 7)}{x(x+7)(x-7)}\]Since there are no common factors to cancel in the numerator and denominator, this is the simplest form.

Key Concepts

Least Common Denominator

When working with rational expressions, a crucial skill is finding the least common denominator (LCD). The LCD is the smallest expression that each denominator can divide without leaving a remainder.
Consider the fractions in the original problem with denominators:

• \( x^2 + 7x \)
• \( x^2 - 7x \)
• \( x^2 - 49 \)

To find the LCD, factor each denominator:

• \( x^2 + 7x = x(x + 7) \)
• \( x^2 - 7x = x(x - 7) \)
• \( x^2 - 49 = (x+7)(x-7) \)

The LCD must include each distinct factor at the highest power present. Therefore, the LCD is \( x(x + 7)(x - 7) \). Having the LCD allows us to combine fractions by rewriting them all with the same denominator.

Factoring in Algebra

Factoring in algebra involves breaking down an expression into products of simpler expressions. This technique is essential when working with rational expressions to simplify them or find a common denominator.
For example, given the quadratic expressions in the denominators in our problem:

• \( x^2 + 7x \) can be factored to \( x(x + 7) \)
• \( x^2 - 7x \) becomes \( x(x - 7) \)
• \( x^2 - 49 = (x + 7)(x - 7) \)

Notice the difference of squares pattern in \( x^2 - 49 \), which directly factors into two conjugates \((x+7)\) and \((x-7)\).
Recognizing these factored forms not only helps in simplification but makes performing operations on rational expressions much easier.

Simplifying Expressions

Simplifying rational expressions often involves combining terms over a common denominator and reducing any redundancy. After rewriting each fraction with the least common denominator, you can then combine and simplify the expression.
For instance, in the step-by-step solution, after finding the LCD, each fraction was rewritten with this denominator:

• \(\frac{x-7}{x(x+7)(x-7)}\)
• \(\frac{2(x+7)}{x(x+7)(x-7)}\)
• \(\frac{5x}{x(x+7)(x-7)}\)

When you subtract these, their numerators are combined:\(x - 7 - 2(x+7) - 5x\) becomes \(-6x - 21\).
Factor the numerator to simplify further: \(-3(2x + 7)\), so the expression simplifies to:\[ \frac{-3(2x + 7)}{x(x+7)(x-7)} \]Be sure to check for any common factors between the numerator and denominator; simplifying can often mean canceling out these shared factors. However, if no further cancellation is possible, as in this problem, the expression is in its simplest form.