Simplifying Rational Expressions

Transcription

1 .3 Simplifying Rational Epressions What are the ecluded values of a rational epression? How can you simplify a rational epression? ACTIVITY: Simplifying a Rational Epression Work with a partner. Sample: You can see that the rational epressions and are equivalent by graphing the related functions y = and y = y y 3 COMMON CORE Rational Epressions In this lesson, you will simplify rational epressions. Learning Standard A.SSE.2 Both functions have the same graph. Match each rational epression with its equivalent rational epression. Use a graphing calculator to check your answers. 2 + b d e A. + B. + C. + D. E Chapter Rational Equations and Functions

2 2 ACTIVITY: Finding Ecluded Values Work with a partner. Are the graphs of y = 2 + and y = + eactly the same? Eplain your reasoning. y 2 y 3 ACTIVITY: Simplifying and Finding Ecluded Values Math Practice Eplain the Meaning What does it mean for a simplified epression to have an ecluded value? Work with a partner. Simplify each rational epression, if possible. Then compare the ecluded value(s) of the original epression with the ecluded value(s) of the simplified epression. d b e. 2 2 f IN YOUR OWN WORDS How can you simplify a rational epression? What are the ecluded values of a rational epression? Include the following rational epressions in your answer. ( + ) b Use what you learned about simplifying rational epressions to complete Eercises 3 5 on page 5. Section.3 Simplifying Rational Epressions 5

3 .3 Lesson Lesson Tutorials Key Vocabulary rational epression, p. 52 simplest form of a rational epression, p. 52 Study Tip You can see why you can divide out common factors by rewriting the epression. ac bc = a c = a b b c b = a A rational epression is an epression that can be written as a fraction whose numerator and denominator are polynomials. Values that make the denominator of the epression zero are ecluded values. Simplifying Rational Epressions Words A rational epression is in simplest form when the numerator and denominator have no common factors ecept. To simplify a rational epression, factor the numerator and denominator and divide out any common factors. Algebra Let a, b, and c be polynomials, Eample where b, c 0. ac bc = a c b c = a b 2( + ) 5( + ) = 2 5 ; EXAMPLE Simplify each rational epression, if possible. State the ecluded value(s). Simplifying Rational Epressions 2 2 = Divide out the common factor. = 2 The ecluded value is = 0. b. n n + 8 Study Tip Make sure you find ecluded values using the original epression. The epression is in simplest form. The ecluded value is n = 8. 3y 2 y(y 7) = 3 y y 2 3 y (y 7) y = 2(y 7) Divide out the common factors. The ecluded values are y = 0 and y = 7. Eercises 3 8 Simplify the rational epression, if possible. State the ecluded value(s).. 5y 3 2y ( + ) m + m(m + 3) 52 Chapter Rational Equations and Functions

4 EXAMPLE 2 Simplify each rational epression, if possible. State the ecluded value(s). Simplifying Rational Epressions z 2 z = ( z)( + z) z (z )( + z) = z (z )( + z) = z = z The ecluded value is z =. Difference of Two Squares Pattern Rewrite z as (z ). Divide out the common factor. b. c 2 + c 2 c 2 c 20 = (c + )(c 3) (c + )(c 5) = c 3 c 5 Factor. Divide out the common factor. The ecluded values are c = and c = 5. EXAMPLE 3 Real-Life Application In general, as the surface area to volume ratio of a substance increases, it reacts faster with other substances. Write and simplify this ratio for a block of ice that has the shape shown. 2 Surface area Volume = 2( 2 ) + (2 2 ) ()(2) 5 = Write an epression. Divide out the common factors. = 5 Eercises 0 5 Simplify the rational epression, if possible. State the ecluded value(s).. 2b + 8 7b a a 2 2a. z 2 z 8 z 7. What is the surface area to volume ratio of a cube-shaped substance with edge length? Section.3 Simplifying Rational Epressions 53

5 .3 Eercises Help with Homework. VOCABULARY Is a rational epression? Eplain REASONING Why is it necessary to state ecluded values of a rational epression? 9+(-)=3 3+(-3)= +(-9)= 9+(-)= Simplify the rational epression, if possible. State the ecluded value(s) w 3 2w 7. t 2 2t(t + ). 5y 3 5y 2 5. n n y 2y 3 9. ERROR ANALYSIS Describe and correct the error in stating the ecluded value(s). 3 2 ( 3) = 3 The ecluded value is = 3. 2 Simplify the rational epression. State the ecluded value(s). 0. 3b + 9 8b z 2z 5 2. a 2 + 2a 9a 3 + 8a 2 3. y 2 y 2 3y 0. n2 + 5n + n 2 + 8n ( + 2)( 5). WRITING Is in simplest form? ( 2)(5 ) Eplain. ( 3) 3 ( 3) 2 7. RECYCLING You hang recycling posters on bulletin boards at your school. Simplify the dimensions of the poster Chapter Rational Equations and Functions

6 Write and simplify a rational epression for the ratio of the perimeter of the figure to its are OPEN-ENDED Write a rational epression whose ecluded values are 3 and WRITING Is 2 equivalent to 2? Justify your answer. + 2 ( ) in. in. h in. 23. PROBLEM SOLVING The candles shown have the same volume. Write and simplify an epression for the height of the cone-shaped candle. 2 in. Sandbo A Sandbo B 2. SANDBOX The area of Sandbo B is square feet greater than the area of Sandbo A. Write and simplify an epression for the width w of Sandbo B. ft (2 3) ft (2 ) ft w ft 25. Find two polynomials whose simplified ratio is + 2 and whose sum is Eplain your reasoning. Graph the function. Is the domain discrete or continuous? (Section 5.2) 2. Input Boes, Output Number of Shoes, y Input Months, Output Height of Plant, y (inches) MULTIPLE CHOICE Consider f () = 2. What is the value of so that f () = 8? (Section 5.) A 2 B C D 7 Section.3 Simplifying Rational Epressions 55