CHAPTER R: Basic Concepts of Algebra R.1 The Real-Number System R.2 Integer Exponents, Scientific Notation, and Order of Operations R.3 Addition, Subtraction, and Multiplication of Polynomials R.4 Factoring R.5 The Basics of Equation Solving R.6 Rational Expressions R.7 Radical Notation and Rational Exponents Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

R.6 Rational Expressions Determine the domain of a rational expression. Simplify rational expressions. Multiply, divide, add, and subtract rational expressions. Simplify complex rational expressions. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Domain of Rational Expressions The domain of an algebraic expression is the set of all real numbers for which the expression is defined. Division by zero is undefined. Example: Find the domain of . Solution: To determine the domain, we factor the denominator. x2 + 3x  4 = (x + 4)(x  1) and set each factor equal to zero. x + 4 = 0 x  1 = 0 x = 4 x = 1 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Simplifying, Multiplying, and Dividing Rational Expressions Solution: Simplify: Solution: Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Another Example Multiply: Solution: Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Another Example Divide: Solution: Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Adding and Subtracting Rational Expressions When rational expressions have the same denominator, we can add or subtract the numerators and retain the common denominator. If the denominators are different, we must find equivalent rational expressions that have a common denominator. To find the least common denominator of rational expressions, factor each denominator and form the product that uses each factor the greatest number of times it occurs in any factorization. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example Subtract: Solution: The LCD is x(x + 2)(x – 2). Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example Add: Solution: The LCD is (3x + 4)(x  1)(x  2). Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Complex Rational Expressions A complex rational expression has rational expressions in its numerator or its denominator or both. To simplify a complex rational expression: Method 1. Find the LCD of all the denominators within the complex rational expression. Then multiply by 1 using the LCD as the numerator and the denominator of the expression for 1. Method 2. First add or subtract, if necessary, to get a single rational expression in the numerator and in the denominator. Then divide by multiplying by the reciprocal of the denominator. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example: Method 1 Simplify: The LCD of the four expressions is x2y2. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example: Method 2 Simplify: Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley