Finding the opposite of a polynomial requires the application of the distributive property. We define the opposite of a polynomial P to be − P. With this understanding, we can cancel any common factors.į ( x ) = ( 2 x − 1 ) ( x + 3 ) ( 2 x + 1 ) ( 2 x − 1 ) = x + 3 2 x + 1Īnswer: f ( x ) = x + 3 2 x + 1, where x ≠ ± 1 2 To find the restrictions, first set the denominator equal to zero and then solve įirst, factor the numerator and denominator.į ( x ) = 2 x 2 + 5 x − 3 4 x 2 − 1 = ( 2 x − 1 ) ( x + 3 ) ( 2 x + 1 ) ( 2 x − 1 )Īny x-value that makes the denominator zero is a restriction. State the restrictions and simplify: f ( x ) = 2 x 2 + 5 x − 3 4 x 2 − 1. Once the restrictions are determined we can cancel factors and obtain an equivalent function as follows: The restrictions to the domain of a rational function are determined by the denominator. S e t - b u i l d e r n o t a t i o n I n t e r v a l n o t a t i o n ( − ∞, 2 ) ∪ ( 2, 3 ) ∪ ( 3, ∞ ) We can express its domain using notation as follows: Therefore, the original function is defined for any real number except 2 and 3. Set each factor in the denominator equal to 0 and solve. To do this, apply the zero-product property. For example, consider the functionį ( x ) = ( x − 1 ) ( x − 3 ) ( x − 2 ) ( x − 3 )īecause rational expressions are undefined when the denominator is 0, we wish to find the values for x that make it 0. We often express the domain of a rational function in terms of its restrictions. are the real numbers for which the expression is not defined. Restrictions The set of real numbers for which a rational function is not defined. consists of all real numbers x except those where the denominator q ( x ) = 0. The domain of a rational function The set of real numbers for which the rational function is defined.

Where p ( x ) and q ( x ) are polynomials and q ( x ) ≠ 0. Rational functions Functions of the form r ( x ) = p ( x ) q ( x ), where p ( x ) and q ( x ) are polynomials and q ( x ) ≠ 0. Identifying Restrictions and Simplifying Rational Functions

Rational expressions