Performing Operations on Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. Understanding how to perform operations on these expressions—like addition, subtraction, multiplication, and division—is crucial for solving algebraic problems. Let’s dive into three practical examples that illustrate these operations in a clear and relatable way.

Example 1: Adding Rational Expressions with Unlike Denominators

Context

Imagine you’re helping a friend plan a party and need to combine two different recipes that require different amounts of ingredients. In this case, we’ll add two rational expressions to find the total amount of one ingredient needed.

Example

Suppose you have the following two expressions representing cups of sugar:

• Expression A: 1/3 cup
• Expression B: 1/4 cup

To add these, we need a common denominator. The least common denominator (LCD) of 3 and 4 is 12.

Now, convert each fraction:

• Expression A: (1/3) = (4/12)
• Expression B: (1/4) = (3/12)

Now, we can add them:

(4/12) + (3/12) = (7/12)

Notes

When adding rational expressions with unlike denominators, always find the least common denominator first. This helps simplify the addition process and ensures accuracy.

Example 2: Subtracting Rational Expressions

Context

Let’s say you’re working on a budget for a school project, and you need to subtract expenses represented by rational expressions. Here, we’ll subtract two fractions to find the remaining budget.

Example

Consider the following expenses:

• Expense A: 5/6 of the budget
• Expense B: 1/2 of the budget

First, identify the least common denominator. For 6 and 2, the LCD is 6. Convert Expense B:

• Expense B: (1/2) = (3/6)

Now, we can subtract:

(5/6) - (3/6) = (2/6)

Simplifying this, we get:
(2/6) = (1/3)

Notes

When subtracting rational expressions, ensure that the denominators are the same before performing the operation. Simplifying the final expression helps in presenting a clearer answer.

Example 3: Multiplying Rational Expressions

Context

You’re working on a science project that involves scaling measurements. In this example, we’ll multiply two rational expressions to find the total area of two rectangular plots of land.

Example

Suppose the dimensions of two rectangles are:

• Rectangle A: (2/3) m wide and (3/5) m long
• Rectangle B: (4/7) m wide and (1/2) m long

To find the area of each rectangle, multiply width by length. For Rectangle A:

Area A = (2/3) * (3/5) = (6/15) = (2/5) m²

For Rectangle B:

Area B = (4/7) * (1/2) = (4/14) = (2/7) m²

Now, let’s find the total area:

Area Total = (2/5) + (2/7)
We need a common denominator for addition here. The LCD of 5 and 7 is 35:

• Area A: (2/5) = (14/35)
• Area B: (2/7) = (10/35)

Now add:
(14/35) + (10/35) = (24/35) m²

Notes

Multiplying rational expressions is straightforward, but when combining areas or other operations afterwards, remember to find a common denominator for addition. This step is crucial for accurate results.

By practicing these examples, you can develop a solid understanding of performing operations on rational expressions, which will be beneficial for tackling more complex algebraic problems in the future.