1. Write three things you already know about expressions. Share your work with a classmate. Did your classmate understand what you wrote?

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1 LESSON 1: RATIONAL EXPONENTS 1. Write three things you already know about epressions. Share your work with a classmate. Did your classmate understand what you wrote?. Write your wonderings about working with epressions.. Write a goal stating what you plan to accomplish in this unit. 4. Based on your previous work, write three things you will do differently during this unit to increase your success. For eample, consider ways you will participate in classroom discussions, your study habits, how you will organize your time, what you will do when you have a question, and so on. Copyright 015 Pearson Education, Inc. 5

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3 LESSON : SIMPLIFYING RADICALS 1. Which number is a simplification of this radical epression? 15 A 10 5 B 15 C 5 5 D 6.5. Assuming that all the variables used in those epressions represent positive numbers, determine which equations are true. There may be more than one correct answer. A a+ a = a B = C t t = t t t D p q r = p q r 1 E a b a = a b. 96 Simplify this radical epression to its simplest form by filling in the empty boes. 4. Simplify this radical epression to its simplest form Simplify this radical epression to its simplest form Simplify this radical epression to its simplest form. 18 Copyright 015 Pearson Education, Inc. 7

4 LESSON : SIMPLIFYING RADICALS 7. Simplify this radical epression to its simplest form A 7 B 7 5. C 49 D Match each radical epression with its simplified form Challenge Problem 9. For what value of n are these two radical epressions equivalent? n 64 6 n and? Show your work and eplain your thinking. Copyright 015 Pearson Education, Inc. 8

5 LESSON : NUMBER SYSTEM 1. You are given the digits and 5. Use both digits to write an integer.. You are given the digits and 5. Use both digits to write a fraction.. You are given the digits and 5. Use both digits to write a sum of two numbers, where the sum is an irrational number. 4. You are given the digits and 5. Use both digits to write an irrational decimal. 5. Which of these epressions are rational numbers? There may be more than one rational epression. A B C.. D 4 + π E Decide whether each number is rational, irrational, or whether you can t tell and place it in the appropriate column. Rational Irrational Can't Tell Copyright 015 Pearson Education, Inc. 9

6 LESSON : NUMBER SYSTEM 7. π 4 is number. A a rational B an irrational Challenge Problem 8. Eplain why non-terminating, repeating decimals are always rational numbers. Use eamples to justify your eplanation. Copyright 015 Pearson Education, Inc. 10

7 LESSON 4: POLYNOMIALS 1. Which justification eplains why the two epressions are equivalent? 7 ( 4y ) = 14 8y A Associative property B Identity property C Commutative property D Distributive property. Which justification eplains why the two epressions are equivalent? 4 7y + + 9y = 4 + 7y + 9y A Associative property B Identity property C Commutative property D Distributive property. Which justification eplains why the two epressions are equivalent? (4 + 7y ) + 9y = (4 + ) 7y + 9y A Associative property B Identity property C Commutative property D Distributive property 4. Which justification eplains why the two epressions are equivalent? y y = (7 4) + y(9 ) A Associative property B Identity property C Commutative property D Distributive property 5. Simplify this polynomial epression Simplify this polynomial epression Copyright 015 Pearson Education, Inc. 11

8 LESSON 4: POLYNOMIALS 7. Look at these two polynomials. A = y B = 7 + y + 4 5y Find the polynomial A B. 8. Simplify this polynomial epression. 9( + 4y ) 7(y ) 9. Simplify this polynomial epression. 0.5(9y + 4) (7y ) 10. Look at these three polynomials. A = (9y y + 4 ) B = 4( + 7y ) (5 + 9y ) C = (9y + 4 7y ) ( + y ) Find the polynomial A B + C. 11. Simplify this polynomial epression. 5( ) 7( ) + ( ) 1. Simplify this polynomial epression. 9( 4y ) 7( 4y + ) + ( y ) 1. Simplify this polynomial epression. (7y 4) + 7(7y 4) 9(7y 4) 14. Simplify this polynomial epression. 4( 7y) 9( 7y) ( + 7y) Challenge Problem 15. What epression do you have to add to 7 for the sum of those two epressions to be 5? Show your work. Copyright 015 Pearson Education, Inc. 1

9 LESSON 5: MULTIPLYING POLYNOMIALS 1. Identify which of the following epressions are equal to ( + ). There may be more than one equivalent epression. A ( ) B C ( + )( + ) D + E 4(1 + ) +. Find the polynomial epression matching each multiplication. ( + 1)(4 4) ( )(1 ) ( 1)( + )( ) ( 5) ( 1)( + 1) ( ) Fill in the coefficients for each term in the trinomial epression that is equal to ( )( ) Simplify this epression. ( + )( + ) 5. Simplify this epression. ( )( ) 6. Simplify this epression. ( 9)( + 4) 7. Simplify this epression. ( + 9)( 4) 8. Simplify this epression. ( + 5)( + 7) Copyright 015 Pearson Education, Inc. 1

10 LESSON 5: MULTIPLYING POLYNOMIALS 9. Simplify this epression. (9 + 7 )( ) 10. Simplify this epression. (9 + 4)( 7 + ) 11. Simplify this epression. (7 + )(4 5) Challenge Problem 1. Two multiplied binomials result in a trinomial: (a + b)(c + d) = R + S + T If a = 1 and c =, what are the values of R, b, and d so that S = 1 and T = 10? Show your work and eplain your thinking. Copyright 015 Pearson Education, Inc. 14

11 LESSON 6: FACTORING 1. Find the factored binomials matching each trinomial epression. b, c, A, and B are all positive numbers. + b + c + b c b c b + c c ( A)( + B) A < B ( A)( + B) A = B ( + A)( + B) ( A)( + B) A > B ( A)( B). Factor this second-degree polynomial. + 8 A ( + )( + 4) B ( )( + 4) C ( + )( 4) D ( )( 4). Find the factored binomials matching each trinomial epression ( + 4)( 6) ( + )( + 4) ( + )( + 9) ( )( 5) ( + 7)( + 8) 4. Find the factored binomials matching each trinomial epression ( + )( + 1) ( )( 5) ( + 8)( + 9) ( )(4 5) ( + )( 7) 5. Which set of factored binomials is equivalent to this trinomial epression? Copyright 015 Pearson Education, Inc. 15

12 LESSON 6: FACTORING 6. Which set of factored binomials is equivalent to this trinomial epression? Factor this second-degree polynomial Factor this second-degree polynomial Factor this second-degree polynomial Factor this second-degree polynomial Factor this second-degree polynomial Factor this second-degree polynomial Challenge Problem 1. The general form of a second-degree polynomial is a + b + c. This polynomial can be factored into the two binomials (m + p)(n + q). a. Determine whether m and n are positive or negative numbers depending on the sign of a. b. Determine whether p and q are positive or negative numbers depending on the signs of b and c. Copyright 015 Pearson Education, Inc. 16

13 LESSON 7: SPECIAL BINOMIALS Complete each sentence to make it true. 1. The polynomial a + ab + b is a. A difference of two squares B square of a sum C square of a difference D sum of a square. The polynomial a ab + b is a. A difference of two squares B square of a sum C square of a difference D sum of a square. The polynomial a b is a. A difference of two squares B square of a sum C square of a difference D sum of a square 4. Look at this epression. ( + ) 9 Factor this epression using one of the special products of binomials. Eplain your thinking and show your work. Copyright 015 Pearson Education, Inc. 17

14 LESSON 7: SPECIAL BINOMIALS 5. Each of these polynomial epressions can be simplified using one of the special products of binomials: Square of a Sum, Square of a Difference, or Difference of Two Squares. Sort each epression to the special product form it belongs to. Square of a Sum Square of a Difference Difference of Two Squares (4 + )(4 ) ( + ) ( 1) Write this epression in polynomial form using the formula of a special product of binomials. Describe which special product you are using and show your work. (4 + ) 7. Write this epression in polynomial form using the formula of a special product of binomials. Describe which special product you are using and show your work. (1 ) 8. Factor this polynomial using the formula of a special product of binomials. Describe which special product you are using and show your work Factor this polynomial using the formula of a special product of binomials. Describe which special product you are using and show your work Miki made an error in her multiplication. Eplain her error and provide the correct answer. ( + )( ) (+)(-) = Copyright 015 Pearson Education, Inc. 18

15 LESSON 7: SPECIAL BINOMIALS Challenge Problem 11. Demonstrate that the epression a ab 4a b + b can be simplified to the product (a b ab)(a b + ab) using special products of binomials. Describe all the steps of your work. Copyright 015 Pearson Education, Inc. 19

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17 LESSON 8: DIVIDING POLYNOMIALS 1. Find the factored binomials matching each trinomial epression Which is the quotient of this epression? + 5 A + 5 B C + 5 D 5. Which equation demonstrates that polynomials are not closed under division? A + 6 = + B = + 1 C 4 = + D = 1 ( ) Copyright 015 Pearson Education, Inc. 1

18 LESSON 8: DIVIDING POLYNOMIALS 4. Find the quotient matching each epression Find the quotient of this epression. Show your work Find the quotient of this epression. Show your work Simplify this epression. 5 ) 8. Find the quotient of this epression. Show your work ( ) 9. Which of these epressions has a quotient equal to ( + )? A B C D Copyright 015 Pearson Education, Inc.

19 LESSON 8: DIVIDING POLYNOMIALS 10. What divisor of 6 16 results in + 4? Challenge Problem 11. m = + q n ( + p) Use your knowledge of polynomial operations to find the values of m, n, p, and q, knowing that p > q. Eplain your thinking. Copyright 015 Pearson Education, Inc.

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21 LESSON 9: OPERATIONS WITH RADICALS 1. Which number is a simplification of this radical epression? A B 1 4 C 5 1 D Find the matching epressions Simplify this radical epression to its simplest form by filling in the empty boes. 4. Find the matching epressions Copyright 015 Pearson Education, Inc. 5

22 LESSON 9: OPERATIONS WITH RADICALS 5. Simplify this radical epression. Show your work. 54 ( + ) 6. Simplify this radical epression. Show your work ( ) 7. Simplify this radical epression. Show your work ( ) 8. Simplify this radical epression. Show your work Simplify this radical epression. Show your work Simplify this radical epression A B 5 4 C D 4 10 Challenge Problem n n 11. a b ma b b What should be the values of m and n for this epression to be equal to ab. Show your work and eplain your thinking. Copyright 015 Pearson Education, Inc. 6

23 LESSON 10: SOLVING RADICAL EQUATIONS 1. Which answer solves this equation? + = 4 A = B = 16 9 C = 4 D = 16. What is the missing number in this equation so that = 9? = + A = B = 0 C = 14 D =. Select the epressions that have a valid solution. There may be more than one epression. A = B + 7 = 4 1 C 1 = D 1 = ( + ) E = Solve this radical epression. Show your work. Use substitution to check your answers. 7 = + 5. Solve each radical epression. Show your work. Use substitution to check your answers. + = 7 Copyright 015 Pearson Education, Inc. 7

24 LESSON 10: SOLVING RADICAL EQUATIONS 6. Solve each radical epression. Show your work. Use substitution to check your answers. 7 = 7. Solve each radical epression. Show your work. Use substitution to check your answers. 1= 7 8. Solve each radical epression. Show your work. Use substitution to check your answers. = 7 9. Solve each radical epression. Show your work. Use substitution to check your answers. = Challenge Problem 10. Remember how the distance formula uses the Pythagorean theorem to find the distance from one point ( 1, y 1 ) to another (, y ). 1 1 d = ( ) + ( y y ) Eplain why, irrespectively of the value of the coordinates of the two points (even negative values), it will always be possible to find the distance between the two points (i.e., there cannot be an impossible radical value). Copyright 015 Pearson Education, Inc. 8

25 LESSON 11: PUTTING IT TOGETHER 1. Read through your Self Check and think about your work in this unit. Write three things you have learned about working with epressions. Share your work with a classmate. Does your classmate understand what you wrote?. Use your notes from class and your thoughts about the unit to review the concepts and properties encountered in the unit. For each concept and property, include a description and one or more eamples. Concept or Property positive integer eponent Description For any nonzero number a and a positive integer n: a n = a a a a (a occurs n times) Eamples Review these concepts and properties:. Review the notes you took during the lessons about epressions. Add any additional ideas you have about this topic to your notes. If you are still confused about a topic, make sure to research your questions and add more information to your notes. Ask for help from a classmate, review the related lessons, look at the resources in the Concept Corner, talk with your teacher, and so on, to help you clear up any confusion. 4. Complete any eercises from this unit you have not finished. Copyright 015 Pearson Education, Inc. 9

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