Addition of Rational Expressions

Definition

Addition of rational expressions is the process of combining two or more rational expressions, which are expressions that can be written in the form of a fraction with a polynomial in the numerator and a polynomial in the denominator. This operation is essential in the context of topics 8.3 and 8.4, which focus on adding and subtracting rational expressions with common denominators and unlike denominators, respectively.

5 Must Know Facts For Your Next Test

1. To add rational expressions with a common denominator, the numerators are added, and the common denominator is retained.
2. When adding rational expressions with unlike denominators, the denominators must first be converted to a least common denominator (LCD) before the numerators can be added.
3. The LCD is the least common multiple of the denominators of the given rational expressions.
4. Multiplying the numerator and denominator of each rational expression by appropriate factors to obtain the LCD is a key step in adding rational expressions with unlike denominators.
5. Simplifying the resulting expression by factoring or canceling common factors is an important final step in the addition of rational expressions.

Review Questions

• Explain the process of adding rational expressions with a common denominator.
  • To add rational expressions with a common denominator, the numerators are simply added together, and the common denominator is retained. For example, to add $\frac{2x}{3x-1}$ and $\frac{5x}{3x-1}$, the numerators are added: $\frac{2x}{3x-1} + \frac{5x}{3x-1} = \frac{2x + 5x}{3x-1} = \frac{7x}{3x-1}$. This process works because the denominators are the same, allowing the fractions to be combined directly.
• Describe the steps involved in adding rational expressions with unlike denominators.
  • When adding rational expressions with unlike denominators, the first step is to find the least common denominator (LCD) of the expressions. This is done by finding the least common multiple of the denominators. Next, each rational expression is rewritten with the LCD as the denominator by multiplying the numerator and denominator by the appropriate factor. Finally, the numerators are added, and the resulting expression is simplified by factoring or canceling common factors. For example, to add $\frac{2x}{3x-1}$ and $\frac{5}{x-1}$, the LCD is $(3x-1)(x-1)$, and the expressions are rewritten as $\frac{2x(x-1)}{(3x-1)(x-1)}$ and $\frac{5(3x-1)}{(3x-1)(x-1)}$, respectively. The numerators are then added: $\frac{2x(x-1) + 5(3x-1)}{(3x-1)(x-1)}$.
• Analyze the importance of simplifying the result when adding rational expressions.
  • Simplifying the result of adding rational expressions is crucial because it ensures the expression is in its most concise and easily understandable form. This may involve factoring the numerator, canceling common factors between the numerator and denominator, or reducing the fraction to its simplest terms. Simplifying the expression not only makes it more visually appealing but also helps to reveal any underlying patterns or relationships within the expression. Additionally, a simplified expression is often easier to manipulate and use in further mathematical operations, making it an essential step in the process of adding rational expressions.

Related terms

Rational Expression: A rational expression is a quotient of two polynomials, where the denominator is not equal to zero.

Common Denominator: A common denominator is a denominator that is shared by two or more rational expressions, allowing for their addition or subtraction.

Unlike Denominators: Unlike denominators refer to rational expressions that have different denominators, requiring a process of finding a least common denominator before addition or subtraction can be performed.