Chapter 5 Self-Assessment

Transcription

1 Chapter 5 Self-Assessment. BLM 5 1 Concept BEFORE DURING (What I can do) AFTER (Proof that I can do this) 5.1 I can multiply binomials. I can multiply trinomials. I can explain how multiplication of binomials is related to area. I can explain how multiplication of binomials is related to the multiplication of two-digit numbers. 5. I can determine prime factors of whole numbers. I can determine the greatest common factors of whole numbers. I can determine the least common multiples of whole numbers. I can write polynomials in factored form. I can apply understanding of factors and multiples to solve problems.

2 . BLM 5 1 (continued) Concept BEFORE DURING (What I can do) AFTER (Proof that I can do this) 5.3 I can develop strategies for factoring trinomials. I can explain the relationship between multiplication and factoring. 5.4 I can factor the difference of squares. I can factor perfect squares.

3 Chapter 5 Prerequisite Skills Show all your work.. BLM 5 1. For each expression, identify the number of terms whether it is a monomial, binomial, trinomial, or polynomial a) 3p b) (3x)(5y) c) h + h 3 d) x 4x + 6. What expression is represented by each set of algebra tiles? Shaded tiles are positive and white tiles are negative. a) b) c) 3. Determine each product. Use a model if necessary. a) ( x)(4.5x) 3t 7 b) ( 14t ) c) ( 0.5s ) s 4 6. A square has side lengths of s. A circle has a diameter that is the same length as the sides of the square. What is the ratio of the areas of the two shapes? 7. Expand. a) (x)(3x 1) b) ( 4k + 1)( 5k) p c) (6x) x d) (3.6p 1.) Divide. (10 b 8 b) a) b 3.9m 1.3 b) 1.3 4h + h c) h m 9. The area of a trapezoid is given by the 1 formula A = ab ( 1+ b), where a is the altitude and b 1 and b are the bases of the trapezoid. What is an expression for the area of the following trapezoid? 4. Determine each quotient. Use a model if necessary. a) 15p 3p b) 8.4n.1n 16.8xy c) ( 4x ) 10. A rectangle has an area of 15v 1v square units. The width of the rectangle is 3v units. What is the length of the rectangle? 5. A rectangle is five times as long as it is wide. If the area of the rectangle is cm, what are its dimensions? Use a diagram to help you.

4 Chapter 5 Warm-Up. BLM 5 3 Section 5.1 Warm-Up 1. Evaluate each expression for x = and y = 3. a) (x + y)(x y) b) x + 5xy 7y. For each expression, multiply the monomial by the polynomial. a) 3x(x y + 5) b) y(5y 8) 4. a) A ruler is 6 cm in length. A piece x cm in length breaks off. Write an expression for the length that is left. b) The radius of a circle is y cm. What is an expression for the diameter of the circle? 5. Write an expression to represent the area of the figure. 3. Simplify each expression. a) (x 5x + 9) + (x + 10x 1) b) (5x + 7xy 4) (8x xy + 3) Section 5. Warm-Up 1. Write each number as a product of prime numbers. a) 7 b) 100. List all the factors of each number. a) 7 b) a) List all the factors of 4. b) List all the factors of 40. c) What is the greatest common factor of 4 and 40? 5. Expand. a) (3x )(x 5) b) 6x(x + 6x 11) 3. List the first five multiples of each number. a) 7 b) 100

5 . BLM 5 3 (continued) Section 5.3 Warm-Up 1. Expand. a) (3x 5)(x + 4) b) (x + 4y)(x 5y). Factor out the greatest common factor. a) 3x + 9x b) 8xy 6y 3. Factor by grouping. a) x(x 5) + (x 5) b) x(x + y) + 5y(x + y) 4. Write all the pairs of integers that multiply to a) 1 b) 7 c) 7 5. a) Write all the pairs of integers that multiply to 6. b) Which pair in part a) adds to 1? c) Which pair adds to 5? Section 5.4 Warm-Up 1. Expand each expression. a) (x 5)(x + 5) b) (x + 4)(x 4). Multiply. a) (x + 5)(x + 5) b) (x 4)(x 4) 3. Multiply. a) (x + 3)(x 3) b) (6x 7)(6x + 7) 4. Expand each expression. a) (x + 3) b) (6x 7) 5. a) What does it mean to factor x + 6x + 9? b) Factor x + 6x + 9. c) Explain how you could check your answer.

6 Chapter 5 Unit Project Section 5.1. BLM Sketch an area model or an algebraic model to represent each multiplication. Use specific polynomials for each multiplication. Label your diagrams. Then, write the result of each multiplication as an equation. a) (monomial)(binomial) b) (binomial)(binomial) c) (binomial)(trinomial). Use an arrangement of algebra tiles to show combining like terms of polynomials. Arrange them artistically. Use the style of Piet Mondrian s paintings, shown here and on page 04 in the student resource. Write the corresponding algebraic equation that summarizes your result. Section Use algebra tiles or area models to show the following relationships. Create a poster displaying your models. a) the relationship between a monomial multiplied by a binomial and common factoring b) the relationship between a binomial multiplied by a binomial and factoring a trinomial of the form ax + bx + c, where a, b, and c are integers 4. a) Use algebra tiles to create a model of a polynomial of your choice. b) Create a piece of art that includes your polynomial in some way. Your artwork may be a drawing, painting, sculpture, or other form of your choice. Section Use models or diagrams to show what happens to the middle terms when you multiply two factors that result in a difference of squares. Include at least two specific examples. 6. a) Use models or diagrams to show the squaring of a binomial. Include at least two specific examples. b) Create a rule for squaring any binomial. Show how your rule relates to your models or diagrams.

7 Section 5.1 Extra Practice. BLM Multiply using algebra tiles. a) (x + )(x + 4) b) (x + 1)(x ). a) What product does the algebra tile model show? b) What are the dimensions of the model? 3. Multiply using the distributive property. a) (x 3)(x 6) b) (y + 10)(y 5) c) (x + 3)(x 4) d) (5 3a)(4 + a) e) 3(x y)(x + y) 4. Multiply using the distributive property. a) (x 5)(x + 5) b) (m + 10)(m 10) c) (x + 3)(x 3) d) (4 3a)(4 + 3a) e) 5(x y)(x + y) 6. Use the distributive property to determine each product. a) x(x + x 1) b) 3a(a + 3a 5) c) (x + )(x x + 5) d) (a 3)(3a + 5a ) e) (x + x 1)(x x + 1) 7. Multiply. Then, combine like terms. a) (x + )(x 3) (x 4) b) (x 1)(x ) + (x + 1)(x + ) c) (a 3) + (a + 4)(a 3) d) (y + z)(y + 10z) (y 5z) e) (x + 3) 4x(x + 4)(3x 1) 8. Multiply. a) (x 3y) (x + y) b) (x + 3) (x 3) c) (x ) (x + 1) d) (x 3) 3 e) (y + 4) 3 9. Write an expression to represent the area of the figure. 5. Use the distributive property to determine each product. a) (x + 4) b) (x 7) c) (6 + y) d) (x + 5y) e) (a + 3b)

8 Section 5. Extra Practice 1. What is the greatest common factor (GCF) of each set of numbers? a) 8 and 98 b) 43 and 16 c) 19 and 16 d) 90, 105, and 165 e) 48, 10, and 168. Determine the least common multiple (LCM) of each set of numbers. a) 1 and 6 b) 9 and 36 c) 6 and 15 d) 4, 5, and 1 e) 16, 0, and Determine the GCF of each set of terms. a) 15x 4 and 5x y b) 4xy and 8xy c) ax and bx d) 18y 4, 9y 3, and 7y e) πxr, πxr, and πxh 4. Factor each polynomial, if possible. a) 5x + 35 b) 4x + 3 c) 14x 8y d) 6x + 4x e) 3x + 9xy + 6xz 6. Factor each polynomial. a) 8x + 3y 3 b) 10a + 5a 5a 3 c) 4abc 6ab + 8bc d) 1x y + 3xy 3 15x 3 y e) 9πx 6xy + 1πxy. BLM Write each expression in factored form, if possible. a) x(y + 1) + 4(y + 1) b) 3x(a + b) y(a + b) c) 4y(y + 3) + (y + 3) d) 5a(x + 1) + 3(x 1) e) 3y(x 5) 4(5 x) 8. Factor by grouping. a) 5x + 15y + mx + 3my b) xy + 4x + 5y + 0 c) 3ab 3ac + b bc d) 5y + 3 6x + 10xy e) x + xz + 6xy + 3yz 9. Write an expression in factored form to represent the area of each shaded region. a) 5. Identify each missing factor. a) 3ax + 3 ay = ( )( x+ y) b) x xy = ( x )( ) c) 5ab 10 ab = (5 ab) ( ) d) x x 3 x = ( ) x x e) 3x 3 x y + 6 xy = ( x )( ) ( 3) b)

9 Section 5.3 Extra Practice. BLM Identify two integers with the given product and sum. a) product = 1, sum = 13 b) product = 34, sum = 19 c) product = 33, sum = 8 d) product = 0, sum = 1 e) product = 54, sum = 15. Factor, if possible. a) x + 8x + 15 b) x + 5x + 6 c) x + 11x + 8 d) m + 7m + 10 e) y + 4y Factor, if possible. a) x 13x + 4 b) x 18x + 81 c) x x 0 d) x + 5x 6 e) x x Factor each trinomial. a) x + 9xy + 14y b) x 8xy + 16y c) x 8xy + 15y d) m + 7mn 8n e) a 6ab 7b 5. Factor each trinomial. First check for a GCF. a) 4x + 4xy + 36y b) x 6x + 7 c) 5x 5xy 30y d) 3x 48x 165 e) 3x 30x Factor. a) x + 13x + 15 b) 3x + 11xy 4y c) 7a 47a + 30 d) 10y + 9y + e) 1x 8x Factor. First check for a GCF. a) 1x 6x 10 b) 18x 3x 36 c) 75y 10y + 48 d) 1x 15xy 18xy e) 40x y 36xy 36y 3 8. Determine two values of b so that each trinomial can be factored. a) x + bx + 10 b) x + bx + 8 c) x bx + 1 d) m + 6m + b e) y + 5y + b 9. Determine two values of k so that each trinomial can be factored. a) x + kx + 5 b) 3x + kx + c) x + kx 15 d) 0m + 3m + k e) 6y + 17y + k

10 Section 5.4 Extra Practice. BLM Determine each product. a) (x + 14)(x 14) b) (a 7)(a + 7) c) (11x + 1)(11x 1) d) (5y 9)( 5y + 9) e) (x + 3)( x 3). What is each product? a) (y + 10) b) (8 m) c) (a 5k) d) 4(3x y) e) (x + 5) 3. Determine the missing values. a) ( ) 4p 5 = ( p) b) 16x 9 = ( ) ( ) c) y 144 = ( y )( y + ) d) 9n 1= ( 3n + )( 3n ) 4 e) x 49 = ( x 7)( ) 4. What are the missing values? a) ( ) x + 10x + 5 = x + b) ( ) p + p = + p c) y 8 y + = ( y 4) d) x = ( x 11) e) 0 w + w = (10 w) 5. Factor each binomial, if possible. a) x 144 b) a 9b c) 5x y d) h + 64 e) 36 a b 6. Factor each trinomial, if possible. a) x + 14x + 49 b) y 40y c) a + a d) 64a 48ab + 9b e) 16x 56xy + 49y 7. Factor completely. a) 16x 4y b) 9x 3 36x c) 7a d) 100ab 5a e) x Factor completely. a) y 4 10y + 5 b) x 4 x + 1 c) 100a 100ab + 5b d) x x y + 00xy e) y y Factor completely. a) (x + 4) 5 b) (a 5) 36 c) 100 (p + 8) d) (x + ) (x ) e) x (y + z) 10. Identify two values of n so that each polynomial will be a perfect square trinomial. Then, factor. a) x + nx + 64 b) y + ny c) 4a + na + 5 d) 9x + nxy + 16y e) 5x + nx + 11

11 Chapter 5 Test. BLM 5 9 Multiple Choice For #1 to 5, select the best answer. 1. What binomial product does the area diagram represent? A (x 1)(x + 3) B (x 3)(x + 1) C (3x + 1)(x ) D (x + )(3x 1). Two students were asked to model the multiplication of two numbers. Their work is shown. Dolores (78)(80) = (80 )(80 + ) = 80 = = 6396 Frank 49 = (50 1) = 50 (50 1) = = 399 Which of the following statements is true? A Both students have a correct procedure. B Neither Dolores nor Frank has a correct procedure. C Frank has an error and Dolores does not have an error. D Dolores has an error and Frank does not have an error. 3. Devin was asked to multiply the expressions 4x 1 and x 5. His work is shown. (4x 1)(x 5) Step 1 = 4x(x 5) (x 5) Step = 8x 0x x 5 Step 3 = 8x x 5 Step 4 Devin verified his answer and realized he had made an error. In which Step did he make his first error? A Step 1 B Step C Step 3 D Step 4 4. Carly wanted to factor the expression x + 5. Which of the following statements is true? A x + 5 = (x + 5) B x + 5 = (x + 5)(x 5) C x + 5 = (x + 5)(x + 5) D Carly cannot factor x + 5 over the integers. 5. Which of the following expressions represents the factors of 3x 17x + 10? A (x )(x 15) B (x 5)(3x ) C (3x )(x 15) D (x 5)(x ) Short Answer 6. a) Draw a diagram to model the product of (x 3)(x + 1). b) Multiply and then combine like terms. 7. Determine the product and then combine like terms. a) (y + 3)(y + 8) b) (5c 9)(4c 1) c) (7a 6y) d) (t 4)(3t 5t + 7)

12 8. Determine the GCF of the terms in each polynomial. a) 4x 3 3x 40x 4 b) 5r s 3 (r + 3) 4rs (r + 3) 9. Factor each expression fully. a) 1a bc 3a b + 4a 3 b 3 b) x 7x 30 c) x 16xy + 64y d) x 5 Extended Response 10. The volume of a rectangular prism can be expressed as 60x x + 45x.. BLM 5 9 (continued) 11. The side length of square A can be expressed as (3x + ) cm. The area of rectangle B is equal to the area of square A increased by (6x + 5) cm. a) Write an expression in fully factored form to represent the area of square A. b) What expression in fully factored form represents the area of rectangle B? c) If the expression (x + 1) cm represents the width of rectangle B, what expression represents its length? a) Determine a possible set of expressions for the length, width, and height of the rectangular prism. b) Verify the product you arrived at in part a) by showing the multiplication of the factors. c) Determine the volume of the prism if x = cm.

13 BLM 5 10 Chapter 5 BLM Answers BLM 5 Chapter 5 Prerequisite Skills 1. a) 1 term, monomial b) 1 term, monomial c) 3 terms, trinomial d) 3 terms, trinomial. a) 3x 3 b) x + 4x c) 3x x a) 9x b) 6t s c) 8 4. a) 5p b) 4n c) 4.y 5. Since x = 50, the dimensions of the rectangle are 50 cm by 50 cm. square 4 circle π 6. = or = circle π square 4 7. a) 6x x b) 0k 5k c) 4x 1x d) 1.p 0.4p 8. a) 5b 4 b) 3m + m c) 4h x + 6x 10. 5v 4 BLM 5 3 Chapter 5 Warm-Up Section a) 5 b) 89. a) 3x 3xy + 15x b) 10y + 16y 3. a) 4x 9 b) 3x + 8xy 7 4. a) 6 x b) y 5. 40x 3x Section a) ()()()(3)(3) b) ()()(5)(5). a) 1,, 3, 4, 6, 8, 9, 1, 18, 4, 36, 7 b) 1,, 4, 5, 10, 0, 5, 50, a) 7, 144, 16, 88, 360 b) 100, 00, 300, 400, a) 1,, 3, 4, 6, 8, 1, 4 b) 1,, 4, 5, 8, 10, 0, 40 c) 8 5. a) 3x 17x + 10 b) 6x x 66x Section a) 3x + 7x 0 b) x + 3xy 0y. a) 3x(x + 3) b) y(4x 3y) 3. a) (x + )(x 5) b) (x + 5y)(x + y) 4. a) (1)(1); ()(6); (3)(4); ( 1)( 1); ( )( 6); ( 3)( 4) b) (1)(7); ( 1)( 7) c) (1)( 7); ( 1)(7) 5. a) (1)( 6); ( 1)(6); ()( 3); ( )(3) b) 3 and c) 6 and 1 Section a) x 5 b) x 16. a) x + 10x + 5 b) x 8x a) 4x 9 b) 36x a) 4x + 1x + 9 b) 36x 84x a) Example: It means rewriting an expression as a product of two binomials. b) (x + 3)(x + 3) or (x + 3) c) by expanding BLM 5 5 Section 5.1 Extra Practice 1. a) x + 6x +8 b) x x. a) (x + 3)(x + 1) = x + 4x + 3 b) length is x + 3; width is x a) x 9x + 18 b) y + 5y 50 c) x 5x 1 d) 0 7a 3a e) 3x 3xy 6y 4. a) x 5 b) m 100 c) 4x 9 d) 16 9a e) 0x 5y 5. a) x + 8x + 16 b) x 14x + 49 c) y + y d) 4x + 0xy + 5y e) 8a + 4ab + 18b 6. a) x 3 + x x b) 3a 3 + 9a 15a c) x 3 + x + 10 d) 6a 3 + a 19a + 6 e) x 4 4x + 4x 1 7. a) x x b) x + 4 c) 3a a 3 d) yz 5z e) 1x 3 44x + 18x a) x 3 11x y + 1xy + 9y 3 b) x 3 + 3x 9x 7 c) x 4 x 3 3x + 4x + 4 d) x 3 9x + 7x 7 e) y 3 +1y + 48y x + 10x + 4 BLM 5 6 Section 5. Extra Practice 1. a) 14 b) 81 c) 4 d) 15 e) 4. a) 156 b) 36 c) 30 d) 60 e) a) 5x b) 8xy c) x d) 9y e) πx 4. a) 5(x + 7) b) not possible c) (7x 4y) d) 6x(x + 4) e) 3x(1 + 3y + z) 5. a) = 3a b) = x y c) = 5 b d) = 3x e) = 3x xy + 6y 6. a) 8(x + 4y 3 ) b) 5a( + a 5a ) c) b(1ac 3a + 4c) d) 3xy( 4xy + y 5x ) e) 3x(3πx y + 4πy ) 7. a) (y + 1)(x + 4) b) (a + b)(3x y) c) (y + 3)(4y + 1) d) not possible e) (x 5)(3y + 4) 8. a) (x + 3y)(5 + m) b) (y + 4)(x + 5) c) (b c)(3a + b) d) (5y 3)(x 1) e) (x + z)(x + 3y) 9. a) 3x(5x + y) b) x(πx 3y) BLM 5 7 Section 5.3 Extra Practice 1. a) 1, 1 b), 17 c) 11, 3 d) 5, 4 e) 6, 9. a) (x + 3)(x + 5) b) (x + 3)(x + ) c) (x + 4)(x + 7) d) (m + )(m + 5) e) (y + 1) 3. a) (x 6)(x 7) b) (x 9) c) (x + 4)(x 5) d) (x 1)(x + 6) e) not possible 4. a) (x + y)(x + 7y) b) (x 4y) c) (x 3y)(x 5y) d) (m + 8n)(m n) e) (a 7b)(a + b) 5. a) 4(x + 3y) b) (x 4)(x 9) c) 5(x 3y)(x + y) d) 3(x + 11)(x + 5) e) 3(x 7)(x 3) 6. a) (x + 3)(x +5) b) (3x y)(x + 4y)

14 c) (7a 5)(a 6) d) (y + 1)(5y + ) e) (x 3)(6x + 5) 7. a) (x 5)(3x + 1) b) 3(3x + 4)(x 3) c) 3(5y 4) d) 3x(4 + 3y)(1 y) e) 4y(5x + 3y)(x 3y) 8. Look for two values for each. a) 7, 11, 7, 11 b) 6, 9, 6, 9 c) 7, 8, 13, 7, 8, 13 d) 5, 8, 9 e) 4, 6 9. Look for two values for each. a) 7, 11, 7, 11 b) 5, 7, 5, 7 c) 1, 7, 13, 9, 1, 7, 13, 9 d) 3, 6 e) 5, 7, 10, 11, 1 BLM 5 9 Chapter 5 Test 1. A. B 3. C 4. D 5. B 6. a) BLM 5 10 (continued) BLM 5 8 Section 5.4 Extra Practice 1. a) x 196 b) 4a 49 c) 11x 1 d) 5y 81 e) x 4 9. a) y + 0y b) 64 16m + m c) 4a 0ak + 5k d) 36x 4xy + 4y e) x x a) = 5 b) (4x) (3) c) (y 1)(y + 1) d) (3n + 1)(3n 1) e) = x a) = 5 b) = 15 c) = 16 d) = x e) = a) (x 1)(x + 1) b) (a 3b)(a + 3b) c) (5x y)(5x + y) d) not possible e) (6 ab)(6 + ab) 6. a) (x + 7) b) (y 0) c) (6 + a) d) (8a 3b) e) (4x 7y) 7. a) 4(x y)(x + y) b) 9x(x )(x + ) c) 3(3a 7)(3a + 7) d) 5a(b 1)(b + 1) e) (x 3)(x + 3)(x + 9) 8. a) (y 5) b) (x 1) (x + 1) c) 5(a b) d) x(x + 10y) e) (y + 9) 9. a) (x 1)(x + 9) b) (a 11)(a + 1) c) ( p)(18 + p) d) 8x e) (x y z)(x + y + z) 10. a) Example: n = 16; (x + 8) or n = 16; (x 8) b) Example: n = 4; (y + 1) or n = 4; (y 1) c) Example: n = 0; (a + 5) or n = 0; (a 5) d) Example: n = 4; (3x + 4y) or n = 4; (3x 4y) e) Example: n = 110; (5x + 11) or n = 110; (5x 11) b) x 5x 3 7. a) y + 11y + 4 b) 0c 41c + 9 c) 49a 84ay + 36y d) 3t 3 17t + 7t 8 8. a) 8x b) rs (r + 3) 9. a) 3ab(7a ab + 8a b ) b) (x + 3)(x 10) c) (x 8y) d) (x 15)(x + 15) 10. a) Look for one set of expressions: 3x; x + 5; 10x + 3 x; 6x + 15; 10x + 3 x; x + 5; 30x + 9 b) Example: 3x(x + 5)(10x + 3) = 3x(0x + 6x + 50x + 15) = 3x(0x + 56x + 15) = 60x + 168x + 45) c) 14 cm 11. a) (3x + ) b) 9(x + 1) c) 9x + 9