Algebraic Fractions: Addition, Subtraction, and Multiplication

Algebraic Fractions: Addition and Subtraction

In mathematics, algebraic fractions are expressions containing at least one variable. They are typically given as ratios between two expressions involving the same variable, such as x or n. The ability to manipulate these fractions is crucial for solving various mathematical problems, particularly those related to algebra and pre-calculus. This article will demonstrate how to add and subtract algebraic fractions.

Understanding Algebraic Fractions

Before we dive into the process of combining algebraic fractions, it's essential to clarify a few concepts. First, let's examine the structure of m/n, where m represents the numerator and n denotes the denominator. In algebraic fractions, m and n consist of variables, constants, or combinations of both. Instead of emphasizing the arithmetic operation being performed (such as addition or multiplication), we focus on the relationship between the numerator and the denominator.

For example, consider the algebraic fraction (a + b)/(c + d). Here, a, b, c, and d represent either variables or constants. The two expressions inside the parentheses () are treated as single entities, allowing us to view the fraction as e/(f + g), where e represents the entire numerator and f + g indicates the denominator.

Combining Algebraic Fractions: Addition and Subtraction

To add or subtract algebraic fractions, we first need to ensure that their denominators are equal. If not, we must find a common denominator and convert both fractions to have this shared denominator. This process is similar to adding or subtracting rational numbers with unlike denominators.

For example, let's consider two algebraic fractions: (2x)/(3y) and (4x - 5)/(6xy). To combine these fractions, we first find a common multiple of 3, 6, x, and y. In this case, the least common multiple (LCM) is 12x^2y^2. We then multiply each fraction by 12x^2y^2/12x^2y^2 to obtain the same denominator for both fractions:

(2x)/(3y) * (12x^2y^2/12x^2y^2) = 24x^2y/(36xy^2)
(4x - 5)/(6xy) * (12x^2y^2/12x^2y^2) = 48x^2(4x - 5)/(720xy^3)

Now that both fractions share the same denominator, we can add or subtract them by combining their numerators:

• For addition: 24x^2y + 48x^2(4x - 5) = 72x^3 - 192x^2y
• For subtraction: 24x^2y - 48x^2(4x - 5) = 72x^3 - 192x^2y

Multiplying Algebraic Fractions

While not directly related to the topic of this article, it's worth mentioning that multiplying algebraic fractions follows a similar process as multiplication in regular arithmetic. We simply multiply the corresponding terms within each fraction and then simplify if possible. No common denominator is required for multiplication.

For example, consider the two algebraic fractions (a + b)/(c + d) and (e + f)/(g + h). To find the product, we multiply like terms together:

((a + b)(e + f)) / ((c + d)(g + h)) = (ae + af + be + bf) / (ce + cg + de + dg)

As mentioned earlier, the main goal when adding or subtracting algebraic fractions is to ensure they have the same denominator. This allows us to combine the numerators and perform the desired operation without any complications.