Problem 46 - Precalculus: Functions and Graphs

Question

Simplify the expression. $$4+\frac{2}{u}-\frac{3 u}{u+5}$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{u^2 + 22u + 10}{u(u+5)} \).

Step 1: Identify the Least Common Denominator (LCD)

To simplify the expression, we first identify the least common denominator for all the terms. The terms have denominators of 1, \( u \), and \( u + 5 \). The LCD is \( u(u+5) \).

Step 2: Rewrite Each Term with the LCD

We will rewrite each term with the least common denominator. This involves adjusting each term to have \( u(u+5) \) as the denominator: 1. The term \( 4 \) can be expressed as \( \frac{4u(u+5)}{u(u+5)} \).2. The term \( \frac{2}{u} \) becomes \( \frac{2(u+5)}{u(u+5)} \).3. The term \( \frac{3u}{u+5} \) becomes \( \frac{3u^2}{u(u+5)} \).

Step 3: Combine the Terms into a Single Fraction

Now that each term is expressed with the least common denominator, combine them into a single fraction:\[ \frac{4u(u+5) + 2(u+5) - 3u^2}{u(u+5)} \].

Step 4: Simplify the Numerator

Distribute and combine like terms in the numerator:- Distribute the \(4u\): \(4u^2 + 20u\).- Distribute the \(2\): \(2u + 10\).Now, the numerator becomes:\[ 4u^2 + 20u + 2u + 10 - 3u^2 \].Combine like terms:\[ (4u^2 - 3u^2) + (20u + 2u) + 10 = u^2 + 22u + 10 \].

Step 5: Write the Simplified Expression

The simplified form of the expression is: \[ \frac{u^2 + 22u + 10}{u(u+5)} \].

Key Concepts

Understanding the Least Common Denominator

When working with algebraic expressions involving fractions, finding the Least Common Denominator (LCD) is crucial for simplifying expressions effectively. The least common denominator is the smallest expression that can be a common denominator for all the individual denominators in a set of fractions. It helps in converting different fractions into equivalent fractions that can be easily added or subtracted.

In this case, consider fractions with different denominators, such as single numbers, variables, and polynomial terms. For the original exercise, the denominators are 1, \( u \), and \( u + 5 \). To find the LCD, we look for a common multiple of these denominators. This involves multiplying the individual factors: \[ \text{LCD} = u(u+5). \]

This LCD allows us to rewrite each part of the algebraic expression with a common denominator, setting the stage for combining all terms into a single fraction.

Process of Combining Like Terms

Combining like terms is an important step in simplifying algebraic expressions. Like terms are terms that have the same variables raised to the same power. The coefficients of these terms can be combined to simplify the expression.

When dealing with the expression given in the original problem, once all terms are rewritten with the same denominator, the top part of the fraction (numerator) needs to be carefully handled. Distributing constants across terms, then identifying and combining like terms, reduces the complexity of the expression.

• For example, collect all terms involving \( u^2 \), \( u \), and numeric constants separately.
• In the original exercise's numerator: \(4u(u+5) + 2(u+5) - 3u^2\) simplifies through distribution to \(4u^2 + 20u + 2u + 10 - 3u^2\).
• Combine to achieve: \(4u^2 - 3u^2 = u^2\) and \(20u + 2u = 22u\), leading to the simplified expression \( u^2 + 22u + 10 \).

By following these steps, you efficiently manage any like terms, streamlining the entire process of simplifying fractions.

Simplifying with Fraction Operations

Simplifying fractions with different denominators is made up of specific operations that help consolidate fractions into simpler forms. Understanding fraction operations in algebra involves more than elementary arithmetic. Here, it's all about aligning your fractions using a common base, thanks to your LCD.

Once fractions have a common denominator, adding or subtracting them is straightforward. Merge all terms over that LCD. For instance, return to our expression:

• Align fractions using the LCD: \(\frac{4u^2 + 20u + 2u + 10 - 3u^2}{u(u+5)}\).
• Combine all numerators since their denominators are now the same.
• Finally, the numerator, after combining like terms, gives a clean simplification: \(\frac{u^2 + 22u + 10}{u(u+5)}\).

This process highlights the beauty of simplification: ease of understanding and proficiency in problem-solving are clarion results of simplifying fraction operations in algebra.