Rational Expressions

Definition

Rational expressions are algebraic expressions that can be written as a ratio of two polynomial functions. They are used to represent and manipulate complex fractions, which are fractions with variables in the numerator and/or denominator.

5 Must Know Facts For Your Next Test

1. Rational expressions can be used to represent and solve a variety of mathematical problems, including those involving rates, ratios, and proportions.
2. The rules for operating with rational expressions (addition, subtraction, multiplication, and division) are similar to the rules for operating with fractions, but they involve manipulating polynomials instead of whole numbers.
3. Simplifying complex rational expressions often requires factoring the numerator and denominator to identify and cancel common factors.
4. Dividing radical expressions can be accomplished by rewriting the expressions as rational expressions and then applying the rules for dividing rational expressions.
5. The degree of a rational expression is determined by the highest degree of the polynomials in the numerator and denominator.

Review Questions

• Explain the process of simplifying a complex rational expression, and provide an example.
  • To simplify a complex rational expression, you need to first identify the common factors in the numerator and denominator, and then cancel them out. This can be done by factoring the numerator and denominator and then canceling any common factors. For example, to simplify the expression $\frac{x^2 - 4x + 3}{x - 1}$, you would first factor the numerator to get $\frac{(x - 3)(x - 1)}{x - 1}$. Then, you can cancel the common factor of $(x - 1)$ in the numerator and denominator, leaving you with the simplified expression $x - 3$.
• Describe the relationship between rational expressions and radical expressions, and explain how to divide radical expressions using rational expressions.
  • Rational expressions and radical expressions are related in that they both involve the manipulation of algebraic expressions with variables. To divide radical expressions, you can rewrite them as rational expressions by rationalizing the denominator. For example, to divide $\sqrt{x}$ by $\sqrt{y}$, you can rewrite it as $\frac{\sqrt{x}}{\sqrt{y}} = \frac{\sqrt{x}}{\sqrt{y}} \cdot \frac{\sqrt{y}}{\sqrt{y}} = \frac{x}{y}$. This allows you to apply the rules for dividing rational expressions to simplify the expression.
• Analyze the role of rational expressions in solving problems involving rates, ratios, and proportions, and provide an example of how to use rational expressions in this context.
  • Rational expressions are particularly useful in solving problems involving rates, ratios, and proportions. These types of problems often involve fractions with variables in the numerator and/or denominator, which can be represented and manipulated using rational expressions. For example, if you are given the rate of $\frac{5}{3}$ miles per hour and you want to find the time it takes to travel a distance of $d$ miles, you can set up the proportion $\frac{5}{3} = \frac{d}{t}$, which can be rearranged to solve for $t$ using rational expression manipulation: $t = \frac{3d}{5}$. This demonstrates how rational expressions can be used to model and solve real-world problems involving rates, ratios, and proportions.

Related terms

Polynomial:

A polynomial is an algebraic expression consisting of variables and coefficients, where the variables are raised to non-negative integer powers.

Complex Fraction: A complex fraction is a fraction where the numerator and/or denominator are also fractions.

Simplify:

Simplifying a rational expression involves reducing the fraction to its lowest terms by factoring the numerator and denominator and canceling common factors.