Name: Date: Problem Set: Find the value of these expressions for the specified replacements of a, b, and c.

Transcription

1 Rational Epressions KEY Module: Objective: Evaluating Rational Epressions To practice evaluating a rational epression for a given set of values Name: Date: Fill in the blanks. Use one of the words in parentheses when choices are given. An number that can be written as a ratio of two integers is a rational (rational, valid) number. Can polnomials be classified as rational epressions if the are written in the form q p where q? es (es or no) The set of all fractional numbers is sometimes called the set of rational numbers. How can ou represent,, and. as rational epressions? 0 Problem Set: Find the value of these epressions for the specified replacements of a, b, and c. a b - c a 6 5b -8 a b 9 b 6 c b -5 a b c 0 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document maaah_k. of

2 Rational Epressions Evaluate these epressions for a and c. Show our work. ( a c) 7 c a (5) a 5ac c a c 9 5(6) 5 8 () ac (c a) 8 (6) ( ) (0) 5 8 a c ac c 9 6 () (9) 9 6 Reflection: Eplain the rules for rational epressions and for rational numbers as if ou were the teacher describing them to a student for the first time. a A rational number is an number that can be written in the form b when b isn't zero. 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document maaah_k. of

3 Rational Epressions Module: Restrictions on Rational Epressions KEY Objective: To practice identifing nonpermissible values for the variables in a rational epression Name: Date: Fill in the blanks. Determine whether each statement below is true or false. Write the correct response in the blank. In a rational epression, an number ma serve as the denominator. false When a denominator is zero, the epression is undefined. true In a rational epression, the variables ma not be replaced b numbers that make the denominator equal to 0. true In some epressions ou must solve the quadratic equation in the denominator to find nonpermissible replacements for variables. true Problem Set: Find the nonpermissible replacement for the variables in these epressions s ( ) 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document maabh_k. of

4 Etension Activit: Rational Epressions Evaluate 8 5 and show our work when: undefined Reflection: Eplain how to identif nonpermissible values for the variables in a rational epression. Use this epression to find the nonpermissible replacement for in our eplanation: 6 Set the denominator equal to zero, and solve the equation to find nonpermissible values. For the above, is not permissible. 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document maabh_k. of

5 Rational Epressions Module: Equivalent Forms of Rational Epressions KEY Objective: To practice identifing rational epressions that are equivalent Name: Date: Fill in the blanks. Use one of the words in parentheses when choices are given. When two fractions are both representations of the same number, the are equivalent fractions. (equivalent, nonequivalent) The rule states that if ou multipl or divide the numerator and denominator b the same number, ou alwas get an equivalent fraction. Cross multipling works with rational epressions. true (true, false) You can alwas check to see whether epressions are equivalent b choosing replacements for the variables in each epression. true (true, false) Problem Set: In each problem, tell whether the rational epressions are equivalent or nonequivalent. equivalent ab ab ab ab equivalent b b nonequivalent b 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document maach_k. of

6 Rational Epressions rst 5rt s 5 equivalent nonequivalent nonequivalent z z s z 5 z ( t ) s t s equivalent nonequivalent nonequivalent nonequivalent equivalent Reflection: Eplain the rule about equivalent fractions in our own words. Use an eample as part of our eplanation. Equivalent fractions are fractions that are equal, but written in different was PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document maach_k. of

7 Module: Simplifing Rational Epressions Rational Epressions KEY Objective: To practice simplifing rational epressions Name: Date: Fill in the blanks. Use one of the words in parentheses when choices are given. The first step in simplifing a rational epression is to factor the numerator and the denominator. You cannot cancel terms found within parentheses when simplifing an epression. false (true, false) To simplif an epression ou must cancel out the common factor. true (true, false) All rational epressions can be simplified. false (true, false) Problem Set: Simplif these epressions.. ab b. ( ) 8ab ( ). 7 z z 7. ( )( ) ( )( ) 5. b b 6b 9 b ( ) ( )( ) 5 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document maadh_k. of

8 Rational Epressions 7. ( a b) a b ( a b) ( a b) s 0s 7 t 5 s t ( t )( t ) 0. t t t t. b ( a ) b( a ) b a. 7 0 ( )( 5) 6 ( )( ) 5. 6 ( ) 8 ( ). 6m 8 m mn 6m n 6( m ) (m n)( m ) 6 m n Reflection: Eplain the steps used to simplif rational epressions. Use this epression to illustrate the steps. 6 First, factor out both the numerator and the denominator: ( )( ) ( )( ) Net, cancel out all common factors: ( )( ) ( )( ) The answer is: 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document maadh_k. of

9 Rational Epressions Module: Sum of Rational Epressions, Part KEY Objective: To practice finding the sum of rational epressions with like denominators Name: Date: Fill in the blanks. Determine whether each statement below is true or false. Write the correct answer in the blank. The rules for adding rational epressions are the same as those for adding fractions. true The rule for adding fractions tells us to add the terms in the numerator. true When adding fractions with common denominators, the denominator does not change. true Problem Set: Epress each sum in the simplest form a a a.. 9 t t t ( t )( t ) t t b b b 5 b 5 b b 5 6. a a a a a a 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document maaeh_k. of

10 Rational Epressions 7. s 6s 9 s s s 6s 9 s ( s ) s s ( ) 9 ( )( ) 9. b b 5 b b b b 0. ( ). 5 s s s 5 s 5 s 5 Reflection: Eplain the rule for adding a rational epression with the same denominator. Eplain this as if ou were the teacher eplaining it to a student for the first time. Use an epression to illustrate the rule. First, add the numerators, combining all like terms. Their sum is the numerator of our final sum. Because the denominators are the same, ou don t have to change the denominator in our answer. If there are an common factors in the numerator and denominator, cancel them out, so our answer is in simplest form. 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document maaeh_k. of

11 Rational Epressions Module: Difference of Rational Epressions, Part KEY Objective: To practice subtracting rational epressions with the same denominator Name: Date: Fill in the blanks. Determine whether each statement below is true or false. Write the correct answer in the blank. The rule for subtracting rational epressions is different from the rule for subtracting fractions. false To find the difference of fractions with the same denominator, subtract the numerators. true You alwas combine like terms when subtracting rational epressions. true Problem Set: Epress each difference in simplest form.. 5 b b b. b b b b b b.. ( ) 5. b 5 6. b 5 b 5 ( b 5)( b 5) b 5 b 5 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document maafh_k. of

12 Rational Epressions 7. t 6t t t t t 8. z z z z z( ) z 9. r r r s r s r r s s s s s s s s. ( ) ( ) Etension Activit: Eplain the rule for subtracting fractions that have the same denominator. Illustrate the rule showing a pie shape. - Subtract the numerators and combine like terms. The denominator stas the same. Simplif b canceling an factors common to the numerator and denominator. 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document maafh_k. of

13 Module: Product of Rational Epressions Rational Epressions KEY Objective: To practice finding the product of two rational epressions Name: Date: Fill in the blanks. The rules for multipling rational epressions are the same as those for multipling real numbers or subsets. What are the three steps for multipling rational epressions?. Multipl the numerator.. Multipl the denominator.. Simplif the epression. If the numerator and the denominator have a common factor, the rational epression can and should be simplified. Problem Set: Epress each product in simplest form a b a b a b a b z z 8z 5. 6 ( )( ) 6 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document maagh_k. of

14 6. r r 6r r r Rational Epressions r ( ) ( ) 6 8. z z 8 z ( ) ( z) ( ) ( )( ) ( ) 5 ( ) ( 5) ( )( 5) Etension Activit: Epress the product in simplest form. 8a br a b 96abz r 6ar 7a z 8ab r a br ( b )(7a 8ar (96abz )( b z ) ) ()(96) a br z ()() a br z 9a r z Reflection: Do ou prefer to simplif before or after multipling? Answers will var. Eplain wh. Answers will var. 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document maagh_k. of

15 Module: Quotient of Rational Epressions Rational Epressions KEY Objective: To practice finding the quotient of two rational epressions Name: Date: Fill in the blanks. The rules for dividing rational epressions are the same as those for dividing fractions. What are the three steps for finding the quotient of two rational epressions?. Flip the divisor and multipl.. Multipl the numerators to get the numerator of the solution. Then, multipl the denominators to get the denominator of the solution.. Check to see if the epression needs to be simplified. Finish the rule: a b c d a d b c Problem Set: Epress each quotient in simplest form a b a 5a 5a b 5a b b b. r 5 r 5 r 5 r 5 r 5 r 5 r 5 r z z z z 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document maahh_k. of

16 Rational Epressions 5. ( )( ) or ( )( )( ) r r 6. r s rs r rs r s r s r ( )( ) 6 ( )( ) b b 9 b b 7 t 7t t t b b b( b ) b b 9 b( b )( b ) ( b ) 7t t 7t t t 7t 7( t ) ( t ) 0. ( ) ( ) ( ) Etension Activit: Circle the epressions that are in simplest form. If the epression is not in simplest form, simplif it on the space provided. 7 z z z ab 9 p 7 r s s _r s Reflection: What process do ou follow for finding the quotient of two rational epressions? Work through an eample to demonstrate these steps. First, flip one of the two rational epressions so that ou have its reciprocal. Net, multipl the numerators to get the numerator of the product, and multipl the denominators to get the denominators of the product. Simplif our product b combining an like terms and canceling out an factors common to the numerator and the denominator. (Eamples will var.) 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document maahh_k. of

17 Rational Epressions Module: Common Denominators of Rational Epressions KEY Objective: To practice finding the least common denominator of two rational epressions Name: Date: Fill in the blanks. Finding the least common denominator of two rational epressions is similar to finding the least common denominator of two fractions What are the steps for finding the least common denominator?. Factor each denominator completel.. Multipl all the different prime factors. For each factor, use the greatest eponent. To find the sum or difference of two rational epressions with unlike denominators, it is often helpful to use the least common denominator. Problem Set: Find the least common denominator for each pair of rational epressions ab 5 9b c 9ab c 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document maaih_k. of

18 . 7 7 Rational Epressions n 0 n n 8 or ( )( ) ( )( )( ) or ( )( ) or n 8 Etension Activit: Draw a line from each pair of epressions on the left to its least common denominator on the right ( )( )( ) 9 6 ( )( )( ) 6 8 ( )( ) 8 ( )()( ) Reflection: Eplain the process ou use to find the least common denominator. Find the different factors of each denominator. Multipl these together to find the lowest common denominator. 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document maaih_k. of

19 Rational Epressions Module: Sum of Rational Epressions, Part KEY Objective: To practice finding the sum of rational epressions with different denominators Name: Date: Fill in the blanks. Use one of the words in parentheses when choices are given. The first step in finding the sum of two rational epressions is to find the least common denominator. This sum is in the simplest form. 5m n m( m n) true (true, false) In the eample 8 5 what are the different prime factors? In the eample b what is the least common denominator? b c c b c Problem Set: Find the sum. Show our work... 9 ( 9 ) PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document maajh_k. of

20 Rational Epressions. n n n 6 n )( n ) ( )( ) n n n n n n ( n ) ( n ) ( n ) n (. r st 6 5 rst r 6rst 0 6rst r 0 6rst Reflection: Eplain how to find the sum of two rational epressions with unlike denominators as if ou were a teacher eplaining it to a student for the first time. Use this problem in our eplanation: m n m ) First, find the lowest common denominator. Because these epressions onl_ have factors of one and themselves, ou can multipl them together to get the lowest common denominator: m(mn). ) Now, to keep equivalent rational epressions, multipl each numerator b whatever its respective denominator was multiplied b. So, the numerator becomes (m), and the numerator becomes (mn). ) Finall, add the numerators, but keep the new denominator the same. Simplif b combining like terms and looking for an common factors that can be cancelled. m m n ( ) ( ) m m n 5m n m n m ( m n)( m) m( m n) m mn m mn 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document maajh_k. of

21 Rational Epressions Module: Difference of Rational Epressions, Part KEY Objective: To practice finding the difference of rational epressions with different denominators Name: Date: Fill in the blanks. Use one of the words in parentheses when choices are given. When ou form a product of all the different prime factors, for each factor use its greatest eponent. (smallest, greatest) When finding the difference of rational epressions with unlike denominators, be sure our answer is in simplest terms. In the problem 6 what is the least common denominator? ( )( ) Problem Set: Find the difference. Show our work.. b c ac ab b c ab c. z z z z 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document maakh_k. of

22 Rational Epressions. b a ab b ab ( 6)( 6) Reflection: What are the easiest and hardest things to remember when finding the difference of rational epressions with unlike denominators? Eplain using this problem: 6 Answers will var. 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document maakh_k. of

23 Module: Review: Rational Epressions Rational Epressions KEY Objective: To review rational epressions Name: Date: Fill in the blanks. Use one of the words in parentheses when choices are given. If the numerator and the denominator have a common factor, the rational epressions can and should be simplified. To find the sum or difference of two rational epressions with unlike denominators, it is often helpful to use the least common denominator. When ou form a product of all the different prime factors, for each factor use its greatest eponent. (smallest, greatest) When adding or subtracting fractions with common denominators, the denominator does not change. true (true, false) Finish the rule: a b c d a d b c This epression is in simplest form. 5m n true (true, false) m( m n) Problem Set: Find the nonpermissible replacement for the variables in these epressions. 8 a 5 z... ( 8) a 5 a 5 0 Tell whether the rational epressions are equivalent or nonequivalent. b c. 5. b 9 c nonequivalent equivalent 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document maalh_k. of c c

24 Epress each sum in simplest form. n nop n nop n 6. ( m ) ( m ) ( m ) 5ab 7ab 6 ab 6 6(7ab ) 7. 7ab r r r 6 r 7r r r r r 6 r 9r 0r 5 Rational Epressions 9. rv stv qr t uv tuv qrs qrtu Epress the difference in simplest form. 6s 7s 6 0. s 6 s 6. v v v 9 v 9 v 6 9. b b 0 b b 5 b 5 b 5 b 5 c 8 ca cb. a b a b Epress the product in simplest form. s s s. p qr s p qr Epress each quotient in simplest form. 6. t r 6 r t 6 t t r r s 0 5s 0 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document maalh_k. of