72 CHAPTER R Review of Basic Concepts R.7 Rational Expressions ■ Rational Expressions ■ LowestTerms of a Rational Expression ■ Multiplication and Division ■ Addition and Subtraction ■ Complex Fractions Rational Expressions The quotient of two polynomials P and Q, with Q≠0, is a rational expression. x + 6 x + 2 , 1x + 621x + 42 1x + 221x + 42 , 2p2 + 7p - 4 5p2 + 20p Rational expressions The domain of a rational expression is the set of real numbers for which the expression is defined. Because the denominator of a fraction cannot be 0, the domain consists of all real numbers except those that make the denominator 0. We find these numbers by setting the denominator equal to 0 and solving the resulting equation. For example, in the rational expression x + 6 x + 2 , the solution to the equation x + 2 = 0 is excluded from the domain. The solution is -2, so the domain is the set of all real numbers x not equal to -2. 5x x ≠-26 Set-builder notation If the denominator of a rational expression contains a product, we determine the domain with the zero-factor property, which states that ab = 0 if and only if a = 0 or b = 0. EXAMPLE 1 Finding the Domain Find the domain of the rational expression. 1x + 621x + 42 1x + 221x + 42 SOLUTION 1x + 221x + 42 = 0 Set the denominator equal to zero. x + 2 = 0 or x + 4 = 0 Zero-factor property x = -2 or x = -4 Solve each equation. The domain is the set of real numbers not equal to -2 or -4, written 5x x ≠-2, -46. S Now Try Exercises 11 and 13. Lowest Terms of a Rational Expression A rational expression is written in lowest terms when the greatest common factor of its numerator and its denominator is 1. We use the following fundamental principle of fractions to write a rational expression in lowest terms by dividing out common factors. Fundamental Principle of Fractions ac bc = a b 1 b 30, c 302
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