Overview
A rational expression has one polynomial in the numerator and one polynomial in the denominator, which means that the numerator is divided by the denominator. However, the denominator cannot be solved to zero, because division by zero is undefined.
Definition
A statement such as (x2 +3)/(y +6) is a rational expression, because it has one polynomial in the numerator and one polynomial in the denominator. The value of x can be set to any rational number, because any number squared will be a positive number. However, the value of y has limits. It cannot be any number that equals -6, because 6-6 equals zero. Similarly, in the rational expression 5/x, x cannot be solved by 0, as 5/0 is undefined. In the rational expression a/ (b-2), b cannot be solved by 2.
Simplest Form
A rational expression is in the simplest form when the only common factors in the numerator and denominator are 1 and -1. Think of fractions in simplest form because their common factors have been eliminated. Expressions such as 6/8 can be simplified to ¾ because 2·3/2·4. The process is similar with rational expressions. For example, the expression (12y +24)/48y can be simplified by first factoring the numerator 12y +24 to 12(y +2). Then the new expression is [12(y +2)]/48y. Then, 12/48 can be factored to ¼. The new expression is (y +2)/4y.
Factoring the Numerator and Denominator
Sometimes both the numerator and denominator of a rational expression need to be factored to simplest terms. Suppose the numerator is in the form 2x2 +x. It can be factored to simplest form as x (2x +1). Suppose the denominator is in the form 3x2 +2x. It can be factored to x (3x +2). The x factors cancel each other out as x/x = 1, so the new expression is (2x +1)/ (3x +2). Similarly, if the numerator is in the form (a2 – 1), it can be factored to (a +1) (a-1). Suppose the denominator of the expression is in the form (2a2-a -1). It can be factored to (a -1) (2a +1) because a·2a equals 2a2, a -2a equals –a, and (-1) (1) equals -1. The new expression is [(a+1)(a-1)]/[(a-1)( 2a+1). The (a-1)/(a-1) factor cancels out to leave a new simplified expression, (a+1)/(2a +1).
Factoring Additive Inverses
Rational expressions such as (x -3)/(3-x) are expressions where the numerator and denominator are additive inverses of each other. To simplify, multiply the numerator by -1 to change the form to -1(x-3). That flips the expression to -1(3-x)/(3-x) or -1. Interested in math tutoring services? Learn more about how we are assisting thousands of students each academic year. SchoolTutoring Academy is the premier educational services company for K-12 and college students. To learn more about how we help parents and students in Ottawa, ON, Canada: visit: Tutoring in Ottawa, ON