Adding And Simplifying Rational Expressions A Step By Step Guide

In the realm of algebra, rational expressions often present a unique challenge, especially when it comes to addition and subtraction. The fundamental principle at play here is the necessity of a common denominator. Just as you can't directly add apples and oranges, you can't directly combine fractions with different denominators. This guide will walk you through the process of adding rational expressions with unlike denominators, using the example provided: $\frac{3}{x} + \frac{5}{3x^2}$.

Understanding Rational Expressions

Before we dive into the solution, it's crucial to understand what rational expressions are. Simply put, a rational expression is a fraction where the numerator and denominator are polynomials. Our given expression, $\frac{3}{x} + \frac{5}{3x^2}$, perfectly fits this definition. The numerators, 3 and 5, are constants (which are also polynomials), and the denominators, $x$ and $3x^2$, are polynomials as well. Adding such expressions requires finding a common ground, which is the least common denominator (LCD).

The Importance of the Least Common Denominator (LCD)

The least common denominator (LCD) is the smallest expression that is divisible by both denominators. Finding the LCD is the cornerstone of adding or subtracting rational expressions. It allows us to rewrite each fraction with a common base, making the addition or subtraction straightforward. In our case, the denominators are $x$ and $3x^2$. To find the LCD, we need to identify the least common multiple of these two expressions.

Finding the LCD: A Detailed Approach

Let's break down the process of finding the LCD for $x$ and $3x^2$:

1. Factor each denominator completely:
   • The first denominator, $x$, is already in its simplest form.
   • The second denominator, $3x^2$, can be thought of as $3 \cdot x \cdot x$.
2. Identify all unique factors:
   • The unique factors are $3$ and $x$.
3. Determine the highest power of each unique factor:
   • The highest power of $3$ is $3^1$ (from $3x^2$).
   • The highest power of $x$ is $x^2$ (from $3x^2$).
4. Multiply the highest powers of all unique factors:
   • LCD = $3 \cdot x^2 = 3x^2$

Therefore, the least common denominator for $\frac{3}{x}$ and $\frac{5}{3x^2}$ is $3x^2$. This means we need to rewrite each fraction with $3x^2$ as the denominator.

Rewriting Fractions with the LCD

Now that we've found the LCD, the next step is to rewrite each fraction with this common denominator. This involves multiplying both the numerator and denominator of each fraction by a suitable factor that will result in the LCD.

Rewriting the First Fraction: $\frac{3}{x}$

To transform the denominator of $\frac{3}{x}$ into $3x^2$, we need to multiply it by $3x$. To maintain the value of the fraction, we must also multiply the numerator by the same factor:

$\frac{3}{x} \cdot \frac{3x}{3x} = \frac{3 \cdot 3x}{x \cdot 3x} = \frac{9x}{3x^2}$

Rewriting the Second Fraction: $\frac{5}{3x^2}$

The second fraction, $\frac{5}{3x^2}$, already has the LCD as its denominator. Therefore, we don't need to change it:

$\frac{5}{3x^2} = \frac{5}{3x^2}$

Adding Fractions with the Common Denominator

With both fractions now having the same denominator, we can proceed with the addition. The rule for adding fractions with a common denominator is simple: add the numerators and keep the denominator the same.

Performing the Addition

Now we can add the rewritten fractions:

$\frac{9x}{3x^2} + \frac{5}{3x^2} = \frac{9x + 5}{3x^2}$

The Result

The sum of the two fractions is $\frac{9x + 5}{3x^2}$. This is a single rational expression, and it represents the combined value of the original two fractions. However, our work isn't quite done yet. The final step is to simplify the result, if possible.

Simplifying the Result

Simplifying a rational expression means reducing it to its simplest form. This involves checking for common factors in the numerator and denominator that can be canceled out. In our case, the expression is $\frac{9x + 5}{3x^2}$.

Checking for Common Factors

• The numerator, $9x + 5$, is a binomial. We need to see if it can be factored. However, there are no common factors between $9x$ and $5$, and the expression cannot be factored further.
• The denominator, $3x^2$, has factors of $3$ and $x^2$ (or $x \cdot x$).

Since there are no common factors between the numerator ($9x + 5$) and the denominator ($3x^2$), the expression is already in its simplest form.

The Final Answer

Therefore, the simplified result of adding $\frac{3}{x}$ and $\frac{5}{3x^2}$ is:

$\frac{9x + 5}{3x^2}$

This is the final answer to our problem.

Key Takeaways

• Adding rational expressions requires a common denominator.
• The least common denominator (LCD) is the smallest expression divisible by all denominators.
• To find the LCD, factor each denominator completely and identify the highest power of each unique factor.
• Rewrite each fraction with the LCD by multiplying both the numerator and denominator by a suitable factor.
• Add the numerators of the fractions with the common denominator.
• Simplify the result by canceling out any common factors between the numerator and denominator.

Practice Makes Perfect

Adding rational expressions can seem daunting at first, but with practice, it becomes a manageable skill. Try working through similar problems, focusing on finding the LCD and rewriting the fractions correctly. Remember to always simplify your final answer.

By mastering the process of adding rational expressions, you'll gain a valuable tool for tackling more complex algebraic problems. The ability to manipulate and combine these expressions is crucial for success in higher-level mathematics.