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11.4 Simplifying Rational Expressions Goals p Simplify a rational expression. p Use rational expressions to find geometric probability. VOCABULARY Rational number A rational number is a number that can be written as the quotient of two integers. Rational expression A fraction whose numerator, denominator, or both numerator and denominator are nonzero polynomials is a rational expression. Simplified rational expression A rational expression is simplified if its numerator and denominator have no factors in common (other than 1). SIMPLIFYING FRACTIONS Let a, b, and c be nonzero numbers. ac apc a bc bpc b Example 1 6x3 3x 12x2 28 4p7 4 Example: 35 5p7 5 Factoring Numerator and Denominator 2p 3pxpxpx 3x (1 4x) 3x (2 x2 ) 3x (1 4x) 2 x2 Factor numerator and denominator. Divide out common factor 3x . Simplified form 1 4x Copyright © McDougal Littell/Houghton Mifflin Company All rights reserved. Chapter 11 • Notetaking Guide 231 Example 2 9 x2 x2 4x 3 Recognizing Opposite Factors ( 3 x )(3 x) Factor numerator and denominator. (x 3)( x 1 ) ( x 3 )(x 3) (x 3)( x 1 ) x3 x1 Factor 1 from ( 3 x ). Divide out common factor x 3 and write simplified form. Checkpoint Simplify the expression if possible. 4x(x 7) 1. 8x2 x7 2x 3(5 x) 2. 12(x 5) 1 4 GEOMETRIC PROBABILITY Region B is contained in Region A. An object tossed onto Region A is equally likely to land on any point in the region. The geometric probability that the object lands in Region B is Area of Region B P . Area of Region A Copyright © McDougal Littell/Houghton Mifflin Company All rights reserved. A B Chapter 11 • Notetaking Guide 232 Example 3 Writing and Using a Rational Model A coin is tossed onto the large rectangular region shown at the right. It is equally likely to land on any point in the region. a. Write a model that gives the probability that the coin will land in the small rectangle. b. Evaluate the model when x 4. 2x 3x 9 x 4x Solution a. P Area of small rectangle Formula for geometric probability Area of large rectangle x( 2x ) Find areas. 4x( 3x 9 ) 2p x p x 4x p 3(x 3) Divide out common factors. x 6(x 3) Simplified form b. To find the probability when x 4, substitute 4 for x in the model. P x 6( x 3 ) 4 6( 4 3 ) 4 2 42 21 2 Answer The probability of landing in the small rectangle is . 21 Checkpoint Refer to Example 3. 3. Find the probability when x 5. 5 48 Copyright © McDougal Littell/Houghton Mifflin Company All rights reserved. Chapter 11 • Notetaking Guide 233