Problem 30 - Algebra A Combined Function

Question

Perform each indicated operation. Simplify if possible. \(\frac{5 x}{6}+\frac{11 x^{2}}{2}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{33x^2 + 5x}{6}\).

Step 1: Identify the Problem

We need to perform the addition of two fractions with algebraic expressions: \(\frac{5x}{6}+\frac{11x^2}{2}\). These fractions have different denominators.

Step 2: Find a Common Denominator

To add fractions, they must have the same denominator. The denominators here are 6 and 2. The least common multiple (LCM) of 6 and 2 is 6. So, we'll use 6 as the common denominator.

Step 3: Adjust the Fractions to Have a Common Denominator

The first fraction \(\frac{5x}{6}\) already has a denominator of 6.The second fraction \(\frac{11x^2}{2}\) needs to be adjusted to have a denominator of 6. Multiply both the numerator and denominator of \(\frac{11x^2}{2}\) by 3 to get:\[\frac{11x^2 \cdot 3}{2 \cdot 3} = \frac{33x^2}{6}\].

Step 4: Perform the Addition of Fractions

Now that both fractions have a common denominator, we can add them by adding the numerators: \[\frac{5x}{6} + \frac{33x^2}{6} = \frac{5x + 33x^2}{6}\].

Step 5: Simplify the Expression

The numerator can be rearranged for clarity: \(33x^2 + 5x\). The expression \(\frac{33x^2 + 5x}{6}\) is already simplified as there are no like terms to combine or factors to cancel.

Key Concepts

Adding Fractions

Adding fractions, especially algebraic ones, can seem daunting at first, but understanding a few key principles can make it much easier. The process is similar whether you're dealing with numbers or algebraic expressions. The key idea is to ensure that both fractions have the same denominator. This allows you to add the numerators directly.
When adding fractions:

• First, identify the denominators of the fractions.
• If they are different, you'll need to find a way to make them the same before you can add anything else.

In our exercise, the fractions are \(\frac{5x}{6}\) and \(\frac{11x^2}{2}\). Before we can add them, we need to ensure they both share one common denominator. This is crucial for combining them into a single expression. Later on, simplification will be your friend as it keeps expressions neat and easy to interpret.

Common Denominator

To successfully add two fractions, the concept of a common denominator is vital. A common denominator refers to the same number at the bottom of each fraction. This allows direct addition of the numerators.
Finding a common denominator involves determining the least common multiple (LCM) of the existing denominators.

• In our example, the denominators are 6 and 2. The LCM of 6 and 2 is 6, which becomes our common denominator.

By having both fractions share the same denominator, we simplify the process of adding them. In the exercise, the fraction \(\frac{11x^2}{2}\) was rewritten to \(\frac{33x^2}{6}\). This was done by multiplying both the numerator and the denominator by 3. Now, both fractions align perfectly with a common denominator of 6, enabling us to simply add their numerators while keeping the denominator constant. This step is crucial in combining the algebraic terms without overlooking the fundamental arithmetic principles that underlie the addition of fractions.

Simplifying Expressions

After adding fractions with a common denominator, the last crucial step is simplifying the expression. Simplification makes an expression more understandable and often reveals insights more easily than a complicated one.
When simplifying an algebraic fraction:

• Start by arranging the terms in the numerator in a logical order, often from highest to lowest degree.
• Check if there are like terms in the numerator that can be combined.

For instance, the resulting expression from adding our fractions is \(\frac{33x^2 + 5x}{6}\). Here, the terms 33\(x^2\) and 5x are already organized but cannot be combined since they aren't like terms.
Finally, investigate if the entire expression can be reduced by factoring.
In our case, there are no common factors between the numerator and denominator, so our expression remains as \(\frac{33x^2 + 5x}{6}\). Simplifying is all about making sure your final answer is the clearest and most reduced form possible, ready for further use or interpretation in more complex problems.