Rational Equation

Definition

A rational equation is an equation that contains one or more rational expressions, where a rational expression is a fraction with a polynomial in the numerator and a non-zero polynomial in the denominator.

5 Must Know Facts For Your Next Test

1. To solve a rational equation, you need to clear the fractions by multiplying both sides of the equation by the least common denominator.
2. Rational equations can have extraneous solutions that do not satisfy the original equation, so it\'s important to check all solutions.
3. The degree of a rational equation is the sum of the degrees of the numerator and denominator polynomials.
4. Rational equations can model real-world situations involving rates, inverse variation, and other relationships between quantities.
5. Graphing a rational equation can help visualize its solutions and identify any asymptotes or other important features.

Review Questions

• Explain the process of clearing fractions to solve a rational equation.
  • To solve a rational equation, you need to clear the fractions by multiplying both sides of the equation by the least common denominator (LCD) of all the rational expressions. This eliminates the denominators and leaves you with a polynomial equation that can be solved using standard techniques, such as factoring or using the quadratic formula. After finding the solutions, you must check each one to ensure it satisfies the original rational equation and does not make the denominator zero, which would result in an extraneous solution.
• Describe how the degree of a rational equation is determined and why it is important.
  • The degree of a rational equation is the sum of the degrees of the numerator and denominator polynomials. This is important because the degree of the equation determines the number and nature of its solutions. For example, a rational equation of degree 1 will have at most one solution, while a rational equation of degree 2 can have up to two solutions. Additionally, the degree of the equation can affect the complexity of the solution process and the behavior of the graph, such as the presence and location of any asymptotes.
• Analyze how rational equations can be used to model real-world situations and the insights they can provide.
  • Rational equations can be used to model a variety of real-world situations involving rates, inverse variation, and other relationships between quantities. For example, a rational equation could be used to model the relationship between the speed of a car and the time it takes to travel a certain distance, or the relationship between the size of a population and the available resources. By solving these rational equations, you can gain insights into the underlying relationships and make predictions about the behavior of the system. Additionally, graphing the rational equation can provide a visual representation of the relationship, which can help identify important features like asymptotes or points of intersection.

Related terms

Rational Expression: A rational expression is a fraction where the numerator and denominator are polynomials, and the denominator is not zero.

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

Extraneous Solution: An extraneous solution is a solution to a rational equation that does not satisfy the original equation, usually because it makes the denominator zero.