Rational Expressions VOCABULARY
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1 11-4 Rational Epressions TEKS FOCUS TEKS (7)(F) Determine the sum, difference, product, and quotient of rational epressions with integral eponents of degree one and of degree two. TEKS (1)(G) Display, eplain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication. Additional TEKS (1)(A), (6)(H) VOCABULARY Rational epression the quotient of two polynomials Simplest form of a rational epression A rational epression is in simplest form when its numerator and denominator have no common divisors. Argument a set of statements put forth to show the truth or falsehood of a mathematical claim Justify To justify is to eplain with logical reasoning. You can justify a mathematical argument. ESSENTIAL UNDERSTANDING You can use much of what you know about multiplying and dividing fractions to multiply and divide rational epressions. roblem 1 TEKS rocess Standard (1)(G) Is there more than one restriction? Yes, before you divided out the common factors, ( + ) was one of the factors of the denominator, so -. Simplifying a Rational Epression What is in simplest form? State any restrictions on the variable ( + )( + 5) Factor the numerator and denominator ( + )( - 5) ( + )( + 5) ( + )( - 5) The simplified form is for 5 and -. The restriction - is not evident from the simplified form, but is needed to prevent the denominator of the original epression from being zero. 476 Lesson 11-4 Rational Epressions
2 roblem Multiplying Rational Epressions What are the following products in simplest form? State any restrictions on the variables. A 3 35y ~ 49y 1z 3 # 49y 35y 1z # 7y # y 3 # z 3 5 # 7y # 7 # 7y 3 # 4z 7 0z The product is 0z 7 for y 0 and z 0. Factor all polynomials. How is multiplying rational epressions like multiplying fractions? To multiply rational epressions, you multiply the numerators and multiply the denominators. B 5 ~ # # # 5 3( - )( + ) - 5 # # 5 3( - )( + ) 3( + ) The product is for 0,, and -. 3( + ) C ~ # ( + 3)( - ) ( + 5)( - 5) # - 5 ( + 3)( + 1) ( + 3)( - ) ( + 5)( - 5) - 5 ( + 3)( + 1) # Factor all polynomials. Factor all polynomials. ( - )( + 5) + 1 ( - )( + 5) The product is + 1 for -3, 5, and -1. earsontexas.com 477
3 roblem 3 Rewriting Quotients of Rational Epressions Rewrite each quotient as a product. Use this to find the quotient of the rational epressions in simplest form. What are the restrictions on the variables? How did you divide fractions without variables? You multiplied by the reciprocal. A 3 4y 6 5z 3 4y, 6 5z 3 # 5z 4y 6 3 # y # 5z # 3 3 # y # 5z # 3 Multiply by the reciprocal. Factor. Cancel common factors. 5z 8y The quotient is 8y 5z. When you identify the restrictions, you need to include both the denominator of the divisor, 5z, and its reciprocal, 6. The restrictions are 0, y 0, and z 0. B , # ( + ) 5( + 1)( - 1) # - 1 7( + )( - ) 7( + ) 5( + 1)( - 1) # - 1 7( + )( - ) 5( + 1) - Multiply by the reciprocal. Factor. Cancel common factors. To identify any restrictions on the variable, determine when - 1, 7-8, and 5 5( + 1) - 5 are equal to 0. The quotient is - for -1, 1, - and. 478 Lesson 11-4 Rational Epressions
4 roblem 4 Dividing Rational Epressions What is the quotient in simplest form? State any 1 restrictions on the variable. How do you start? Think of division as multiplying by the reciprocal. To divide, you multiply by the reciprocal. The epressions may have common factors. So, factor the numerators and denominators # # ( + 1)( 1) ( + 1)( + 1) ( + 5)( ) Factor -1 from ( - ) to get a second ( - ). 1( ) # ( + 1)( 1) ( + 1)( + 1) ( + 5)( ) Divide out common factors. 1( ) ( + 1)( 1) ( + 1)( + 1) ( + 5)( ) Rewrite the remaining factors. 1( 1) ( + 1)( + 5) Identify the restrictions from the denominator of the simplified epression and from any other denominator used. 1, 5, 1, and earsontexas.com 479
5 roblem 5 TEKS rocess Standard (1)(A) Using Rational Epressions to Solve a roblem Construction Your community is building a park. It wants to fence in a play space for toddlers. It wants the maimum area for a given amount of fencing. Which shape, a square or a circle, provides a more efficient use of fencing? One measure of efficiency is the ratio of area fenced to fencing used, or area to perimeter. Which of the two shapes has the greater ratio? How can you compare 16 and 4 without evaluating? Since the numerators are the same, the fraction with the smaller denominator is the larger fraction. Square Circle Area s Define area and perimeter. Area pr erimeter 4s s 4 Area erimeter s ( 4 ) Epress s and r in terms of a common variable,. Write the ratios. Substitute for s and r. erimeter pr r p 16 Since 4p 7 16, a circle provides a more efficient use of fencing. Check Assume 40 ft. The area of the circle is p ( p) ft. Area erimeter pr The area of the square is ( 40 4 ) 100 ft. The area of the circle is greater. p ( p ) 4p ONLINE H O M E W O R K RACTICE and ALICATION EXERCISES Scan page for a Virtual Nerd tutorial video. For additional support when completing your homework, go to earsontexas.com. Simplify each rational epression. State any restrictions on the variables y 15y z z c + 9c 3c Multiply. State any restrictions on the variables # # # a # 15b 9b 14c # # Lesson 11-4 Rational Epressions
6 Divide. State any restrictions on the variables y 18z, 7y y y -, y y - 1 y + 4, , 6y y , y - 5y + 6 y 3, y + 3y y 19. Apply Mathematics (1)(A) A kitchen storage container will have a circular base of radius r and a height of r. The container can be either cylindrical or hemispherical (half a sphere). a. Write and simplify an epression for the ratio of r the volume of the hemispherical container to its surface area (including the base). For a sphere, V 4 3 pr3 and SA 4pr r. b. Write and simplify an epression for the ratio of the volume of the cylindrical container to its surface area (including the bases). c. Compare the ratios of volume to surface area for the two containers. d. Compare the volumes of the two containers. e. Describe how you used these ratios to compare the volumes of the two containers. Which measurement of the containers determines the volumes? 0. Create Representations to Communicate Mathematical Ideas (1)(E) Write three rational epressions that simplify to Apply Mathematics (1)(A) A cereal company wants to use the most efficient packaging for their new product. They are considering a cylindrical-shaped bo and a cube-shaped bo. Compare the ratios of the volume to the surface area of the containers to determine which packaging will be more efficient. Multiply or divide. State any restrictions on the variables , , # # r r 6. a. Eplain Mathematical Ideas (1)(G) Write a simplified epression for the area of the rectangle at the right. b. Which parts of the epression do you analyze to determine the restrictions on a? Eplain. c. State all restrictions on a. 3a 9 a 6 4a 4 a 3 earsontexas.com 481
7 Decide whether the given statement is always, sometimes, or never true. 7. Rational epressions contain eponents. 8. Rational epressions contain logarithms. 9. Rational epressions are undefined for values of the variables that make the denominator Restrictions on variables change when a rational epression is simplified. State any restrictions on the variables. ( - ) 31. ( - 1) - ( y ) 3 - y ( - 1) -1 ( + - 3) 34. a. Use Representations to Communicate Mathematical Ideas (1)(E) Simplify ( n ) - 1 n, where is an integer and n is a positive integer. (Hint: Factor the - 1 numerator.) b. Use the result from part (a). Which part(s) of the epression can you use to show that the value of the epression is always odd? Eplain. a b Use the fact that c a b c d to simplify each rational epression. State any d restrictions on the variables. 8 y 3a 3 b 3 9m + 6n a - b m 35. 6y n 4ab 1m + 8n b - a 5m TEXAS Test ractice 39. Which function is graphed at the right? A. y ( + 4)( - 1)( + ) B. y ( - 4)( - 1)( + ) C. y ( - 4)( + 1)( - ) D. y ( + 4)( + 1)( - ) 40. Which function generates the table of values at the right? F. y log 1 G. y -log H. y log J. y ( 1 ) # Which epression equals ? - 1 A. C. ( - 1)( + 3)( + 1) ( - 1)( + 1)( - 3) + 1 B. D. ( - 1)( + 1)( - 3) ( + 3)( - 1)( + 1) 4. What is the solution of the equation ? 8 y O y Lesson 11-4 Rational Epressions
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