A rational expression is a mathematical expression that can be written as a ratio of two polynomial functions. It represents the quotient of two algebraic expressions, where the denominator is never zero.
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Rational expressions can be simplified by factoring the numerator and denominator and canceling common factors.
The domain of a rational expression is the set of all real numbers, except the values that make the denominator equal to zero.
Rational expressions can be added, subtracted, multiplied, and divided using specific rules and procedures.
Partial fractions are a technique used to decompose a rational expression into a sum of simpler rational expressions.
Rational expressions are often used in calculus, physics, and other scientific fields to model and analyze real-world problems.
Review Questions
Explain the process of simplifying a rational expression by factoring the numerator and denominator.
To simplify a rational expression, you first need to factor both the numerator and denominator. This allows you to identify and cancel any common factors between the numerator and denominator. By canceling out these common factors, you can reduce the rational expression to its simplest form. This process is important because it makes the expression easier to work with and can reveal important properties or relationships within the expression.
Describe the role of the domain in the context of rational expressions.
The domain of a rational expression is the set of all real numbers, except for the values that make the denominator equal to zero. This is because division by zero is undefined, and the rational expression would not be defined for those values. Understanding the domain of a rational expression is crucial, as it determines the range of input values for which the expression is valid and can be evaluated. Identifying the domain is an important step in analyzing and working with rational expressions.
Analyze how the technique of partial fractions is used to decompose a rational expression into simpler components.
Partial fractions is a technique used to decompose a rational expression into a sum of simpler rational expressions. This is particularly useful when working with more complex rational expressions, as it allows you to break them down into manageable pieces that can be more easily manipulated and analyzed. The process of partial fractions involves identifying the factors of the denominator and then using algebraic methods to express the original rational expression as a sum of these simpler fractions. This decomposition can provide valuable insights and facilitate further mathematical operations, such as integration, that may be necessary in the context of the problem.