Problem 48 - Algebra A Combined Function

Question

Perform each indicated operation. Simplify if possible. \(\frac{x-6}{5 x+1}+\frac{6}{(5 x+1)^{2}}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{5x^2 - 29x}{(5x+1)^2}\).

Step 1: Analyze the Denominators

Look at each term within the fraction. The first term is \(\frac{x-6}{5x+1}\) with a denominator of \(5x+1\). The second term is \(\frac{6}{(5x+1)^2}\) with a denominator of \((5x+1)^2\). The common denominator would be \((5x+1)^2\) as it is the higher power of \(5x+1\).

Step 2: Rewrite Each Fraction with Common Denominator

Rewrite the first fraction, \(\frac{x-6}{5x+1}\), using the common denominator \((5x+1)^2\):\[\frac{(x-6)(5x+1)}{(5x+1)^2}\]The second fraction already has the common denominator \((5x+1)^2\).

Step 3: Perform the Addition

Combine the fractions using their common denominator:\[\frac{(x-6)(5x+1)}{(5x+1)^2} + \frac{6}{(5x+1)^2} = \frac{(x-6)(5x+1) + 6}{(5x+1)^2}\]

Step 4: Simplify the Numerator

Expand and simplify the expression in the numerator, \((x-6)(5x+1) + 6\):\[(x-6)(5x+1) = 5x^2 + x - 30x - 6 = 5x^2 - 29x - 6\]Adding the 6:\\[5x^2 - 29x - 6 + 6 = 5x^2 - 29x\]So the numerator becomes \(5x^2 - 29x\).

Step 5: Write the Final Simplified Expression

Substitute the simplified numerator back with the common denominator:\[\frac{5x^2 - 29x}{(5x+1)^2}\]This is the expression for the sum of the two fractions simplified.

Key Concepts

Common Denominator

Understanding the concept of a common denominator is crucial when adding fractions. When dealing with fractions, both parts must share the same denominator before they can be combined. In our example, the denominators given are \(5x+1\) and \((5x+1)^2\). The least common denominator between these terms is \((5x+1)^2\), because it is the larger expression in terms of power.Why do we choose this? Because a common denominator allows you to rewrite each fraction so that they can be expressed in terms of the same base. This makes the task of addition feasible. When the denominators are the same, the numerators can be directly added, aiding in the overall simplification of the expression. It's like finding a common language for both fractions to "speak."

Numerator Simplification

After finding a common denominator and rewriting each term to reflect this, the next step involves simplifying the numerator. This starts by expanding any algebraic expressions within the numerator. Let's break it down further:

• Identify any expressions to expand: For example, the expression \((x - 6)(5x + 1)\).
• Perform the multiplication: Apply distributive property to expand \((x - 6)(5x + 1) = 5x^2 + x - 30x - 6\).
• Simplify: Combine like terms to further reduce the expression to \(5x^2 - 29x - 6\).

This simplification process helps in expressing the sum in the simplest, cleanest form possible, making it more straightforward to understand and solve.

Algebraic Expression

Algebraic expressions include variables, constants, and operations. They form the backbone of our problem. In the context of this fraction addition, algebra helps manage how numbers and variables interact to yield an expression. For our example:

• The terms \(x\), \(6\), and expressions like \(5x + 1\) are pivotal to forming our fractions.
• Knowing how to manipulate these—like multiplying \((x - 6)(5x + 1)\)—requires an understanding of algebraic principles.
• Operations such as addition and multiplication are applied to rearrange and simplify the expression.

Grasping how these expressions work lets you simplify fractions into a comprehensible format, solving them correctly and efficiently.

Fraction Simplification

Once the numerators have been simplified and a common denominator established, the final step is to simplify the entire fraction as much as possible.This involves:

• Ensuring that the numerator and the denominator are as reduced as they can be.
• Checking for any common factors in both parts.
• In our context, ultimate simplification was achieved by reducing \(5x^2 - 29x\) over \((5x+1)^2\) without common factors.

Simplifying fractions in arithmetic and algebra helps in making complex expressions easier to understand. It facilitates better analytical ability and allows for more straightforward solving of problems, ensuring clarity and precision in mathematical assignments.