Subtraction of Rational Expressions

Definition

Subtraction of rational expressions involves finding the difference between two rational expressions, which are fractions with polynomial expressions in the numerator and denominator. This operation is essential in simplifying and manipulating rational expressions, a crucial skill in algebra.

5 Must Know Facts For Your Next Test

1. To subtract rational expressions, the denominators must first be made the same by finding the least common denominator (LCD).
2. Once the LCD is found, the numerators of the rational expressions are subtracted, and the result is placed over the common denominator.
3. Subtracting rational expressions often requires factoring the numerator and denominator to simplify the resulting expression.
4. Subtraction of rational expressions is useful in solving equations, evaluating functions, and simplifying complex algebraic expressions.
5. The process of subtracting rational expressions is similar to subtracting fractions, but the numerators and denominators are polynomial expressions rather than just numbers.

Review Questions

• Explain the steps involved in subtracting two rational expressions.
  • To subtract two rational expressions, 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, the numerators of the expressions are subtracted, and the result is placed over the common denominator. Finally, the expression is simplified by factoring the numerator and denominator and canceling any common factors.
• Describe the role of the least common denominator (LCD) in subtracting rational expressions.
  • The least common denominator (LCD) is crucial in subtracting rational expressions because the denominators must be the same for the subtraction to be performed. The LCD ensures that the denominators are the same, allowing the numerators to be subtracted directly. Finding the LCD requires identifying the prime factorization of each denominator and then selecting the highest power of each prime factor to create the LCD. This step ensures that the resulting expression is in its simplest form.
• Analyze how the process of subtracting rational expressions is similar to and different from subtracting fractions with whole numbers.
  • The process of subtracting rational expressions is similar to subtracting fractions with whole numbers in that the denominators must be the same, and the numerators are then subtracted. However, the key difference is that the numerators and denominators of rational expressions are polynomial expressions, rather than just whole numbers. This means that additional steps, such as factoring the numerator and denominator and canceling common factors, are often necessary to simplify the resulting expression. The use of the least common denominator is also more crucial in subtracting rational expressions to ensure the denominators are compatible for the subtraction operation.

Related terms

Rational Expression:

A rational expression is a fraction where both the numerator and denominator are polynomial expressions.

Least Common Denominator (LCD):

The least common denominator is the smallest common multiple of the denominators of the rational expressions being subtracted.

Simplifying Rational Expressions: The process of reducing a rational expression to its simplest form by factoring the numerator and denominator and canceling common factors.