Problem 14 - Algebra 1: Concepts and Skills

Question

Find the least common denominator of the pair of rational expressions. $$ \frac{x-1}{x-2}, \frac{x-3}{x-4} $$

Step-by-Step Solution

Answer

The least common denominator of the two rational expressions is \((x-2)(x-4)\).

Step 1: Find the denominators

The denominators of the given rational expressions are \(x-2\) and \(x-4\).

Step 2: Calculate the least common denominator

As the two denominators \(x-2\) and \(x-4\) are not the same, the least common denominator can be obtained by simply multiplying the two together. Hence, the least common denominator of the two rational expressions is \((x-2)(x-4)\).

Key Concepts

Rational Expressions

Rational expressions are similar to fractions, but instead of only numbers, they allow for polynomial expressions in the numerator and denominator. Just like fractions, understanding rational expressions is crucial for various mathematical operations. In rational expressions, you can perform operations such as addition, subtraction, multiplication, and division. These operations often require manipulating the expressions to have common denominators. When working with rational expressions, ensure that the denominator never equals zero, as this would make the expression undefined. Finding a common denominator is an important step in simplifying or solving problems involving rational expressions.

Denominators

The denominator of a rational expression is the part of the expression found beneath the fraction line. It indicates how many equal parts the whole is divided into. In rational expressions with polynomial denominators, it's essential to factorize expressions to understand their behavior. For instance, in the expressions \(\frac{x-1}{x-2}\) and \(\frac{x-3}{x-4}\), the denominators are \(x-2\) and \(x-4\), respectively. To handle these expressions effectively, we often need to find a common denominator for the expressions. Having a common denominator simplifies operations like addition or subtraction, enabling us to combine the fractions into a single expression.

Algebra Concepts

Algebra is a branch of mathematics dealing with symbols and rules for manipulating those symbols. When dealing with rational expressions, algebra concepts such as factoring, expanding, and simplifying are frequently used. To solve the original exercise, we use several algebraic steps:

• Factoring: Identifying the factors of numerators and denominators helps in simplifying rational expressions or finding common denominators.
• Polynomial Multiplication: To find the least common denominator, expressions like \((x-2)(x-4)\) are multiplied, expanding if needed to further simplify or analyze expressions.
• Simplification: After expressing rational quantities with a common denominator, algebra principles can help reduce them to their simplest forms.

These concepts are not only fundamental for solving problems but also for understanding the relationships and functions involving rational expressions.