Problem 5 - Algebra A Combined Function

Question

Perform each indicated operation. Simplify if possible. \(\frac{3}{x}+\frac{5}{2 x^{2}}\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{6x + 5}{2x^2} \).

Step 1: Identify the fractions

The expression is made up of two fractions: \( \frac{3}{x} \) and \( \frac{5}{2x^2} \). Our goal is to add these fractions.

Step 2: Find a common denominator

To add fractions, they must have the same denominator. The denominators are \( x \) and \( 2x^2 \). The least common denominator (LCD) of these is \( 2x^2 \).

Step 3: Adjust the fractions

Convert each fraction to have the LCD \( 2x^2 \) as its denominator. Multiply the numerator and denominator of \( \frac{3}{x} \) by \( 2x \) to get \( \frac{6x}{2x^2} \). The fraction \( \frac{5}{2x^2} \) already has the correct denominator.

Step 4: Add the fractions

Now that both fractions have the same denominator, add the numerators: \( \frac{6x + 5}{2x^2} \).

Step 5: Simplify the expression

Check if the numerator can be simplified. In this case, \( 6x + 5 \) cannot be factored further to cancel out with the denominator, so the expression remains \( \frac{6x + 5}{2x^2} \).

Key Concepts

Least Common Denominator

When working with algebraic fractions, just like with numerical fractions, you need a common denominator to add or subtract them. The **least common denominator (LCD)** is the smallest expression that all denominators can divide into without remainder. For this exercise, the fractions are \( \frac{3}{x} \) and \( \frac{5}{2x^2} \), with denominators \( x \) and \( 2x^2 \), respectively. To find the LCD:

• List the distinct variables and constants in the denominators: here we have \( x \) and \( 2 \).
• Take the highest power of each variable present in the denominators: \( 2x^2 \) includes both \( 2 \) and \( x^2 \).
• This gives us the LCD of \( 2x^2 \). All terms in each fraction can now be aligned using this common denominator.

Therefore, multiplying the terms to achieve this LCD allows seamless addition of the fractions.

Adding Fractions

Once the least common denominator is determined, you can focus on transforming each fraction to have this denominator before combining them. For the fractions \( \frac{3}{x} \) and \( \frac{5}{2x^2} \), you need each to share the denominator \( 2x^2 \).Steps to align denominators:

• Multiply both the top (numerator) and bottom (denominator) of \( \frac{3}{x} \) by \( 2x \) to get \( \frac{6x}{2x^2} \). Note that we multiply by \( 2x \) because \( x \cdot 2x = 2x^2 \), which matches our LCD.
• The fraction \( \frac{5}{2x^2} \) already has this common denominator, so no changes are necessary for it.

Combining the Fractions

Simply add the numerators over the common denominator. After adjustment, the expression is \( \frac{6x}{2x^2} + \frac{5}{2x^2} \), and adding these gives \( \frac{6x + 5}{2x^2} \). Always verify that numerators are correctly combined as mistakes can happen.

Simplifying Expressions

Once the fractions are added and combined into a single expression, the next step is **simplifying**. To simplify a fraction, check if the numerator and denominator have common factors that can be cancelled. In our expression \( \frac{6x + 5}{2x^2} \), we aim to simplify the numerator.Steps to check for simplification:

• Examine the numerator and see if it can be factored. In this case, \( 6x + 5 \) is a simple binomial that cannot be factored further.
• Look at both the numerator and the denominator for any common factors. Here, there are none common to both \( 6x + 5 \) and \( 2x^2 \).
• If there are no common factors, the fraction is already in its simplest form.

Keep in mind that simplification is crucial for clearer and more manageable expressions. But in this case, since \( 6x + 5 \) cannot be simplified, the expression stays as is: \( \frac{6x + 5}{2x^2} \).