Problem 42 - Algebra A Combined Function

Question

Perform each indicated operation. Simplify if possible. \(\frac{5}{(x+1)(x+5)}-\frac{2}{(x+5)^{2}}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{3x + 23}{(x+1)(x+5)^2}\).

Step 1: Common Denominator

Identify the common denominator for the fractions. The denominators are \((x+1)(x+5)\) and \((x+5)^2\). The common denominator will be \((x+1)(x+5)^2\).

Step 2: Adjust the First Fraction

Transform the first fraction to have the common denominator: \(\frac{5}{(x+1)(x+5)}\) becomes \(\frac{5(x+5)}{(x+1)(x+5)^2}\). Multiply the numerator by \((x+5)\).

Step 3: Adjust the Second Fraction

Transform the second fraction to have the common denominator: \(\frac{2}{(x+5)^2}\) becomes \(\frac{2(x+1)}{(x+1)(x+5)^2}\). Multiply the numerator by \((x+1)\).

Step 4: Combine the Fractions

Both fractions now have the same denominator, allowing them to be combined: \(\frac{5(x+5) - 2(x+1)}{(x+1)(x+5)^2}\).

Step 5: Simplify the Numerator

Perform the subtraction in the numerator: \((5x + 25) - (2x + 2)\) simplifies to \(3x + 23\). The combined expression becomes \(\frac{3x + 23}{(x+1)(x+5)^2}\).

Step 6: Final Simplification and Solution

Check for any common factors in the simplified fraction. There are no common factors between \(3x + 23\) and \((x+1)(x+5)^2\), so the expression is simplified as \(\frac{3x + 23}{(x+1)(x+5)^2}\).

Key Concepts

Common Denominator

Finding a common denominator is like finding a shared foundation. It allows us to combine fractions into a single expression. Let's explore this concept using our example fractions. We need to identify the least common denominator (LCD) for the denominators

• \((x+1)(x+5)\)
• \((x+5)^2\)

To do this, include all factors from each denominator, ensuring each factor appears as many times as it does in any single denominator. This results in the common denominator:

• \((x+1)(x+5)^2\)

To work with the fractions effectively, each must have this common denominator. Adjusting fractions to have an LCD allows us to perform operations like subtraction, since they have the same "base" to work from.

Fraction Subtraction

Subtracting fractions is straightforward once they share a common denominator. Think of it as subtracting the numerators while keeping the denominator constant. Here’s how it applies to our exercise:First, rewrite each fraction with the common denominator:

• The first fraction, \(\frac{5}{(x+1)(x+5)}\), becomes \(\frac{5(x+5)}{(x+1)(x+5)^2}\). The numerator is adjusted by multiplying it with \((x+5)\).
• The second fraction, \(\frac{2}{(x+5)^2}\), becomes \(\frac{2(x+1)}{(x+1)(x+5)^2}\). This time, multiply the numerator by \((x+1)\).

Now, we can subtract:

• Combine the numerators: \(5(x+5) - 2(x+1)\)
• Keep the common denominator: \((x+1)(x+5)^2\)

Performing the subtraction of the numerators while keeping the denominator consistent ensures clarity and accuracy in problem-solving.

Simplifying Expressions

Simplifying an expression is the art of making it as uncomplicated as possible, without changing its value. After subtracting the numerators, the expression to simplify is \(\frac{(5x + 25) - (2x + 2)}{(x+1)(x+5)^2}\).

• First, simplify the numerator: perform the subtraction to get \(3x + 23\).
• Check for common factors between the numerator and the denominator. If any exist, cancel them out.

In our case, \(3x + 23\) has no common factors with \((x+1)(x+5)^2\). This means the expression is already in its simplest form: \(\frac{3x + 23}{(x+1)(x+5)^2}\).
By simplifying, we make the expression easier to understand and use in further calculations.