Name Period Date. polynomials of the form x ± bx ± c. Use guess and check and logic to factor polynomials of the form 2

Name Period Date POLYNOMIALS Student Packet 3: Factoring Polynomials POLY3 STUDENT PAGES POLY3.1 An Introduction to Factoring Polynomials Understand what it means to factor a polynomial Factor polynomials that have a common monomial factor Factor binomials that fit the difference of two squares pattern POLY3.2 Exploring Factoring Patterns with a Model Understand what it means to factor a polynomial Factor polynomials with a model Multiply polynomials Write polynomials in a conventional form Explore relationships in the coefficients of polynomials written as a sum of terms and polynomials written as the product of factors POLY3.3 Factoring Trinomials Understand what it means to factor a polynomial Apply patterns explored to develop techniques for factoring 2 polynomials of the form x ± bx ± c. Use guess and check and logic to factor polynomials of the form 2 ax ± bx ± c POLY3.4 Vocabulary, Skill Builders, and Review 24 1 9 17 POLY3 SP

WORD BANK (POLY3) Word Phrase Definition or Explanation Example or Picture binomial constant term difference of two squares distributive property factor greatest common factor (GCF) linear term monomial polynomial product quadratic term trinomial POLY3 SP0

3.1 Introduction to Factoring Polynomials AN INTRODUCTION TO FACTORING POLYNOMIALS Ready (Summary) We will learn that factoring can be interpreted as undoing multiplication. We will factor polynomials that have a common monomial factor and binomials that fit the difference of two squares pattern. 1. What are all of the factors of 24? What are all the factors of 36? Go (Warmup) What are all the common factors of 24 and 36? Set (Goals) What is the greatest common factor (GCF) of 24 and 36? 2. What are all the factors of 3x 2? What are all the factors of 6x? What are all the common factors of 3x 2 and 6x? What is the greatest common factor (GCF) of 3x 2 and 6x? Understand what it means to factor a polynomial Factor polynomials that have a common monomial factor Factor binomials that fit the difference of two squares pattern POLY3 SP1

3.1 Introduction to Factoring Polynomials A GARDEN PROBLEM PATTERN The greatest common factor (GCF) of a polynomial refers to the largest monomial that divides into each of the terms of the polynomial without a remainder. Example: The GCF for 3x 2 + 6x is 3x because 3x is the greatest monomial that divides into both 3x 2 and 6x without a remainder. 3x 2 + 6x = 3x(x+2) polynomial as a sum of terms = polynomial as a product of factors Below is a garden pattern diagram, all made of x-pieces and unit pieces. The shaded parts are the planted area and the un-shaded parts are the border area. Garden 1 Garden 2 Sketch Garden 3. 1. Complete the table. Garden # height length (h) base length (b) TOTAL area as a product (b h) TOTAL area as a sum greatest common factor (GCF) 1 3 x + 2 ( + ) + 2 3 4 5 6 2. Under what conditions is the GCF the same as the height? 3. Why is the GCF not always the height? POLY3 SP2

3.1 Introduction to Factoring Polynomials FACTORING POLYNOMIALS: GCF Factoring can be interpreted as the undoing of multiplication. When factoring a polynomial, first look for the greatest common factor (GCF) of all the terms. Then use the distributive property to write the polynomial as the product of the GCF and another polynomial. 1. Factor 24x 36 What is the GCF Make a generic rectangle and determine the missing factors. So, 24x 36 = of the terms 24x and -36? ( ) Factor each polynomial. Illustrate with a generic rectangle. 2. 12x 9 = ( )( ) 12x -9 24x -36 3. 5x + 45 = 4. 3x 2 + 6x 12 = 5. 10x 2 80x = 5x POLY3 SP3

3.1 Introduction to Factoring Polynomials FACTORING POLYNOMIALS: GCF (Continued) Sometimes it is useful to factor out the opposite of the greatest common factor. 6. Factor -2x 2 6x Factoring out the GCF: 2x -2x 2-6x -2x -2x 2 6x = 2x( ) Factoring out the opposite of the GCF: -2x 2-6x -2x 6 = -2x( ) Using a generic rectangle, first factor out the opposite of the GCF, and then the other polynomial factor as well. Then write each polynomial as a product of the opposite of the GCF and another factor. 7. -5x + 45 8. -10x 2 80x 9. -2x 2 + 4x 8 10. 7x 49x 2 11. Look at each of the factored polynomials above. Why do you think it might be useful to write these polynomials using the opposite of the GCF rather than the GCF? POLY3 SP4

3.1 Introduction to Factoring Polynomials PRACTICE WITH GCF Find the greatest common factor for each pair of monomials. 1. 3 and 12 2. 6x and 2x 2 3. 7x and 18 4. 24x and 10 5. x 2 and 2x 6. 2x 2 and 2x 7. Under what conditions is the GCF = 1? Identify the greatest common factor of the terms, and then write the expression as the product of the greatest common factor (GCF) and another factor. expression as a sum (sum of terms) GCF of the terms expression as a product (GCF) (another factor) 1. 4x + 20 4 ( + ) 2. x 2 7x 3. 9x + 6x 2 + 12 4. -27x 45 5. -7y + y 2 6. 3x 3 x 7. -5 25x 8. 4x 2 + 16x 9. -x 3 + 3x 2 + 5x 10. 2x 3 6x 2 + 14x 11. x 2 + 5x 12. 2x + 7 POLY3 SP5

3.1 Introduction to Factoring Polynomials GARDEN PROBLEM PRACTICE Below is a garden pattern diagram, all made of x-pieces and unit pieces. The shaded parts are the planted area and the un-shaded parts are the border area.. Garden 1 Garden 2 Sketch Garden 3 1. Complete the table. Garden # 1 2 3 4 5 6 height length (h) base length (b) TOTAL area as a product (b h) TOTAL area as a sum greatest common factor (GCF) Sandy wants to build a garden where the length is 3 feet shorter than the width. 3. Draw Sandy s garden. 4. Represent the area of the garden as a: a. product of factors b. sum of terms 5. Suppose the width of Sandy s garden is 6 feet. What is the area of his garden? Which form of the expression did you use to make this calculation? Why? POLY3 SP6

3.1 Introduction to Factoring Polynomials THE DIFFERENCE OF TWO SQUARES Multiply using a strategy of your choice. 1. 7(x + 7) = 2. x(x 7) = 3. (x + 7)(x + 7) = 4. (7 x)(7 x) 5. (x + 7)(x 7) = 6. (7 x)(7 + x) 7. The products in problems 5 and 6 follow a special pattern. This pattern is called:. a. Write another polynomial as a product of factors and a sum of terms that fits the pattern you named in problem 7. b. Generalize this pattern using symbols or words. Cross off all binomials that DO NOT fit the difference of two squares pattern. Factor all binomials that fit this pattern. 8. x 2 36 9. x 2 + 9 10. x 2 9x 11. 144 x 2 12. 4y 2 25 13. 16x 2 20 14. 25x 2 16 15. 9x 2 100 16. 4x 9 POLY3 SP7

3.1 Introduction to Factoring Polynomials FACTORING BINOMIALS 1. Give an example and a non-example of a binomial. 2. Give an example of the difference of two squares pattern for factoring binomials. When factoring binomials, first look for a greatest common factor. Then look to see if the binomial fits the difference of two squares pattern. Factor each binomial. If it cannot be factored using techniques you have learned, then write not factorable. 3. x 2 9 4. -6x 2 24 5. 25x 2 + 25 6. 12x 2 18x 7. x 2 25 8. 49 x 2 9. 9x 2 + 16 10. 9 + 81x 11. 2x 2 5y 2 Each binomial below has three factors. Factor each binomial completely. 12. 5x 2 125 13. 4x 2 36 14. x 4 1 POLY3 SP8

3.2 Exploring Factoring Patterns with a Model EXPLORING FACTORING PATTERNS WITH A MODEL Ready (Summary) We will use math pieces and a one-region area model to deepen our understanding of what it means to factor polynomial expressions. We will create rectangles using math pieces and observe patterns in the coefficients of the polynomials they represent. These explorations are intended to help us become more skilled at identifying factoring patterns. 1. Multiply: 3x(x + 5) Use a squares, sticks, and dots diagram. 2. Multiply: (x + 6)(2x + 1) Use a squares, sticks, and dots diagram. Go (Warmup) Set (Goals) Use a generic rectangle. Use a generic rectangle. Understand what it means to factor a polynomial Factor polynomials with a model Multiply polynomials Write polynomials in a conventional form. Explore relationships in the coefficients of polynomials written as a sum of terms and polynomials written as the product of factors Use the distributive property. Use the distributive property. POLY3 SP9

3.2 Exploring Factoring Patterns with a Model FACTORING WITH MATH PIECES Factoring can be interpreted as the undoing of multiplication. To factor using an area model, form a rectangle using pieces that represent the product. Then determine two factors by observing the dimensions of the rectangle. (product)= (factor) (factor) Factor. Use your One-Region Workmat to build each expression as a rectangle. Then record a sketch, and write the side lengths as factors. 1. 3x + 6 = ( )( ) 2. x 2 + 5x + 4 = ( )( ) Hint: use 3 of the x-pieces and 6 units to make a rectangle. What are the side lengths? Multiply the factors. What is the result? Multiply the factors. What is the result? 3. x 2 + 2x = 4. x 2 + 2x+ 1 = Multiply the factors. What is the result? Hint: use one x 2 -piece, 5 x-pieces and 4 units to make a rectangle. What are the side lengths? Multiply the factors. What is the result? 5. Compare problems 3 and 4. Why do the x-pieces have to be placed differently? POLY3 SP10

3.2 Exploring Factoring Patterns with a Model FACTORING WITH MATH PIECES (continued) Factor. Use your One-Region Workmat to build each expression as a rectangle. Then record a sketch below using squares, sticks, and dots and write the side lengths as factors. Check your factors by multiplying. 6. x 2 + 5x = 7. x 2 + 5x + 6 = Hint: Since one x 2 -piece and one x-piece are already placed, four more x-pieces are needed. 8. x 2 + 4x = 9. x 2 + 4x + 3 = 10. x 2 + 3x = 11. x 2 + 3x + 2 = 12. x 2 + 7x = 13. x 2 + 7x + 12 = POLY3 SP11

3.2 Exploring Factoring Patterns with a Model CREATING RECTANGLES Factor. Use your One-Region Workmat to build each expression as a rectangle. Then sketch below using squares, sticks, and dots in conventional form, and write the side lengths as a product of factors. 1. x 2 + 6x = 2. x 2 + 6x + 5 = 3. x 2 + 6x + 8 = 4. Build one more different rectangle using one x 2 -piece, 6 x-pieces, and as many unit pieces as you like. Sketch it below. a. Write this polynomial as a sum of terms. b. Write this polynomial as a product of factors. 5. Are there other unique rectangles that can be built using one x 2 -piece, 6 x-pieces, and as many unit pieces as you like? Explore with Math Pieces and, if there are any, show them in sketches below. If no others exist, explain why not. POLY3 SP12

3.2 Exploring Factoring Patterns with a Model RECTANGLE EXPLORATION 1 1. Create as many rectangles as you can using one x 2 -piece (in the lower left corner) and ten x-pieces. You may use as many units as needed. Make sketches of your rectangles using squares, sticks, and dots. Write each polynomial as a product and as a sum. Use your own paper to create additional diagrams if needed. 2. How do you know you found all the possibilities and you didn t miss any? 3. Look at the constant terms of the polynomial written as a product. a. How do these numbers relate to the constant term when the polynomial is written as a sum? b. How do these numbers relate to the coefficient of the x-term when the polynomial is written as a sum? POLY3 SP13

3.2 Exploring Factoring Patterns with a Model RECTANGLE EXPLORATION 2 1. Create as many rectangles as you can using one x 2 -piece (in the lower left corner) and eight x-pieces. You may use as many unit pieces as needed. Make sketches of your rectangles using squares, sticks, and dots. Write each polynomial as a product and as a sum. Use your own paper to create additional diagrams if needed. 2. Look at the constant terms of the polynomial written as a product. a. How do these numbers relate to the constant term when the polynomial is written as a sum? b. How do these numbers relate to the coefficient of the x-term when the polynomial is written as a sum? POLY3 SP14

3.2 Exploring Factoring Patterns with a Model RECTANGLE EXPLORATION 3 Use Math Pieces to create rectangles. Make sketches on your own paper. Write each polynomial as a product and as a sum. Write down relationships you observe among the coefficients of the terms in the polynomials. Create as many rectangles as you can using: one x 2 -piece, as many x-pieces as needed, and: 1. five unit pieces 2. six unit pieces 3. seven unit pieces 4. eight unit pieces 5. nine unit pieces 6. ten unit pieces 7. How do you know you found all the possibilities and you didn t miss any? 8. Look at the constant terms of the polynomial written as a product. a. How do these numbers relate to the constant term when the polynomial is written as a sum? b. How do these numbers relate to the coefficient of the x-term when the polynomial is written as a sum? POLY3 SP15

3.2 Exploring Factoring Patterns with a Model RECTANGLE EXPLORATION QUESTIONS For the exercises on this page, a factor pair will refer to a pair of whole number factors for a given product. For example, one factor pair for 8 is 1 and 8; another factor pair for 8 is 2 and 4. There are no other factor pairs for 8. 1. How many factors pairs are there for 6? They are. 2. For problem 2 on the previous page (one x 2 -piece, 6 unit pieces, and as many x-pieces as you like), you should have drawn two different rectangles. Redraw your two rectangles A. Write the polynomial as a sum. Circle the constant term. Underline the coefficient of the linear term. B. Write the polynomial as a product of binomials. Circle the constant terms in each factor. C. How are the constant terms in B related to the constant term in A? D. How are the constant terms in B related to the coefficient of the linear term in A? Continue the exploration by answering the questions below, but do not draw rectangles unless you need them. 3. For one x 2 -piece, 12 unit pieces, and as many x-pieces as you like, how many rectangles do you think can be drawn? Explain how you know. 4. Using what you know from observing patterns, try to write all of the polynomials for these rectangles as sums and products. If needed, sketch them (use extra paper). polynomial as sum of terms polynomial as product of factors x 2 + x + 12 (x + )(x + ) POLY3 SP16

3.3 Factoring Trinomials Ready (Summary) We will apply patterns we have explored to develop techniques for factoring simple trinomials. We will use guess and check and other logical techniques to factor more complicated trinomials. Multiply: FACTORING TRINOMIALS Go (Warmup Set (Goals) Understand what it means to factor a polynomial Apply patterns explored to develop techniques for factoring polynomials of the form ax 2 ± bx ± c Use guess and check and logic to factor polynomials of the form ax 2 ± bx ± c 1. (x + 5)(x + 2) 2. (x + 1)(x + 9) 3. (x + 7) 2 4. In each example, how are the constant terms in the original multiplication expression related to the constant term in the product? 5. In each example, how are the constant terms in the original multiplication expression related to the coefficient of the linear term in the product? Use your pattern to multiply each of these polynomials mentally. 6. (x + 8)(x + 3) 7. (x + 12)(x + 5) 8. (x + 15)(x + 2) 9. Explain how you multiplied the binomials in problem 8 above mentally. POLY3 SP17

3.3 Factoring Trinomials Multiply: POLYNOMIAL MULTIPLICATION PATTERNS 1. (x 5)(x + 2) 2. (x + 5)(x 2) 3. (x 5)(x 2) 4. (x 1)(x + 9) 5. (x + 1)(x 9) 6. (x 1)(x 9) 7. In each example, how are the constant terms in the original multiplication expression related to the constant term in the product? 8. In each example, how are the constant terms in the original multiplication expression related to the coefficient of the linear term in the product? Use your pattern to multiply each of these polynomials mentally. 9. (x 8)(x + 3) 10. (x + 12)(x 5) 11. (x 15)(x 2) 12. Explain how you multiplied the binomials in problem 10 above mentally. 13. In the warmup, all the binomials had positive constant terms. On this page, the binomials have positive and negative constant terms. Is your strategy for multiplying these binomials the same or different from the previous page? Explain. POLY3 SP18

3.3 Factoring Trinomials 2 FACTORING TRINOMIALS OF THE FORM x ± bx ± c Recall that factoring can be interpreted as the undoing of multiplication. Factor each polynomial completely. Polynomial as a sum Think Polynomial as a product Check by multiplying Polynomial as a sum Think: Polynomial as a product Check by multiplying Polynomial as a sum: Think: Polynomial as a product: 1. x 2 + 8x + 15 2. 2x 2 20x + 48 Is there a GCF? If so, factor it out first. What two numbers have a product of 15 and a sum of 8? Is there a GCF? If so, factor it out first. What two numbers have a product of and a sum of? (x + )(x + ) 2 (x )(x ) 3. x 2 + 5x 14 4. x 2 6x 27 5. 4x 2 8x 24 6. x 2 13x + 40 Check by multiplying: POLY3 SP19

3.3 Factoring Trinomials TRINOMIAL FACTORING PRACTICE Factor using any method. Look for a GCF first, then factor each polynomial completely. If it cannot be factored with the skills you have learned, write not factorable. 1. x 2 + x 90 2. x 2 18x + 77 3. x 2 + 8x + 15 4. x 2 + 4x + 2 5. x 2 + 2x 24 6. 3x 2 3x + 60 7. x 2 10x 21 8. x 2 6x 20 9. 2x 2 + 12x + 10 10. x 2 10x + 25 11. x 2 + 10x 24 12. x 2 10x 24 In each problem above the coefficient of the square term is positive. Look for other patterns relating the original polynomial and its factors. 13. When the constant term in the trinomial is positive, what is true about the signs in the two binomial factors? 14. When the constant term in the trinomial is negative, what is true about the signs in the two binomial factors? 15. Write (+) and ( ) signs in between the spaces to describe symbolically the patterns you observe. Use problems 1-12 above to guide you. a. x 2 + bx + c ( )( ) b. x 2 bx + c ( )( ) c. x 2 + bx c and x 2 bx c ( )( ) or ( )( ) POLY3 SP20

3.3 Factoring Trinomials 2 FACTORING TRINOMIALS OF THE FORM ax ± bx ± c 1. Vivienne, Celeste, and Lucette are working as a team to try to factor the following polynomial: 2x 2 + 11x + 5 A Vivienne says: I don t see any common factors, so I think it factors into two binomials and the signs in the binomials are positive. Do you agree? Explain. B Celeste says: I think the only factor pair for the first terms in the binomial must be 2x and x. Do you agree? Explain. C Lucette says: I think the only factor pair for the second term in the binomial must be 5 and 1. I don t think there are any other possibilities. Do you agree? Explain. Vivienne wrote this (and she left the blanks there): ( + )( + ) Celeste wrote this (and she left the blanks there): (2x + )(x + ) Lucette wrote these possibilities: (2x + 5)(x + 1) (2x + 1)(x + 5) D Finish the problem by multiplying the possibilities. To save time, multiply the one you think is correct first. Were you correct? Why did you choose that one to try first? Factor completely. Use logical reasoning to help you to create lists of possibilities and to choose which one to multiply first. 2. 2x 2 + 5x 7 3. 3x 2 + 6x + 3 4. 5x 2 54x 11 POLY3 SP21

3.3 Factoring Trinomials 2 FACTORING TRINOMIALS OF THE FORM ax ± bx ± c (continued) 5. Now the team attempts to factor: 3x 2 7x 6 A Vivienne says: I don t see any common factors, but I think it factors into two binomials and the signs are different. Do you agree? Explain. B Celeste says: I think the only factor pair for the first terms in the binomial must be 3x and x. Do you agree? Explain. C Lucette says: I think the only factor pairs in the second part of the binomial must be 2 and 3. Let s write out all the possibilities and check them by multiplying. Do you agree? (Help them finish the problem in the space to the right.) D Multiply the one you think might be correct first. Were you correct? Why did you choose that one first? Vivienne wrote this (and she left the blanks there): ( + )( ) or ( )( + ) Celeste wrote this (and she left the blanks there): (3x + )(x ) or (3x )(x + ) List all of the possibilities that Lucette should have written (hint: there are more than two). Factor completely. Use logical reasoning to help you to create lists of possibilities and to choose which one to multiply first. 6. 2x 2 + 8x + 6 7. 3x 2 + 2x 8 8. 5x 2 4x 12 POLY3 SP22

3.3 Factoring Trinomials FACTORING TRINOMIALS STRATEGIES 1. Give an example of a trinomial and a non-trinomial. Factor each trinomial. First look for the GCF of all the terms. 1. 6x 2 4xy + 2x 2. x 2 13x + 22 3. x 2 + 3x 18 Sometimes a polynomial cannot be factored. If possible, factor each trinomial. If it cannot be factored using techniques you have learned, then write not factorable. 4. x 2 + x + 1 5. 2x 2 + 5y 7z 6. x 2 17x + 30 If a polynomial written as a sum has more than two factors, it may require more than one step to factor it completely. Each of the trinomials below has more than three factors. Try to factor them completely. 7. 7x 2 35x + 42 8. 30x 2 + 10x 20 9. x 3 + 6x 2 + 5x POLY3 SP23

3.4 Vocabulary, Skill Builders, and Review Here are three equations. FOCUS ON VOCABULARY (POLY3) 3x(x + 7) = 3x 2 + 7x (x 7)(x + 7) = x 2 49 2x 2 7x + 3 = (2x 1)(x 3) Use these equations to give one or two examples of a: 1. constant term 2. linear term 3. quadratic term 4. monomial 5. binomial 6. trinomial 7. polynomial 8. product 9. factor 10. difference of two squares 11. greatest common factor 12. polynomial written as a product of factors 13. polynomial written as a sum of terms 14. distributive property POLY3 SP24

3.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 1 1. The slope-intercept form of a linear equation is 2. The standard form of a linear equation is 3. The point-slope form of a linear equation is Write the equations of the lines in the following forms in any order desired. 4. Given: two points on the line are (0, 0) and (5, 4) 5. Given: m = 1 and one 2 point on the line is (-2, -6) 6. Given: the x-intercept is 6 and the y-intercept is -2 a. slope-intercept: a. slope-intercept: a. slope-intercept: b. standard: b. standard: b. standard: c. point-slope: c. point-slope: c. point-slope: Solve each system using algebra. Check by substitution. 7. 5x + 3y = -15 y = 2x + 6 8. 5y 4x = -9 5y = 3x - 7 POLY3 SP25

3.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 2 Graph each inequality. Consider the equation of the boundary line and graph it as a solid or dashed line (slopeintercept form of a line is usually easier to graph). Shade the appropriate half-plane. Test at least one point by substituting it into the inequality. 1. -y < 2(-x 1) 2. y + 3 1 (x + 4) 2 3. Solve for x: 3x 2y = 15 4. Solve for y: 3x 2y = 15 Compute: 5. (-5) 2 6. (-5) 3 7. (-5) -2 8. Marla said that 3 4 + 3 4 = 2 3 4. Is Marla correct? Explain. POLY3 SP26

3.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 3 For each equation, determine if it is true or false. Show work to justify your answer. 1. x 2 x 2 = x 2 + (-x 2 ) 2. -(2x + 3) = -2x + 3 3. (x 3) (x + 4) = x 3 x + 4 4. (2x 2 4) (x + 3) = 2x 2 x 1 5. (x 2) (x + 2) = x 2 x + 2 6. 3(x + 5) + 2(x + 5) = 6(x + 5) 7. Angela said that x means the same thing as 1x. Is she correct? Explain. 8. A Math Pieces garden picture has been started below. Finish the diagram. Then write the dimensions and the area. The shaded pieces ARE part of the garden and should be included in the dimensions. Height (h) = Base (b) = Area = bh = ( )( ) = Write dimensions here POLY3 SP27

3.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 4 Simplify each expression below so that there are no longer parenthesis. Use any method. Look for patterns to help you multiply more efficiently if possible. 1. 4(5x + 8) 2. -4(5x 8) 3. (x 8)(x + 8) 4. (x + 2) 2 5. x(x + 2) + 2(x + 2) 6. -(x 5) + 7x(x + 3) 7. 7(-8 x 2 ) 8. -(x 6) 2(x + 1) 9. (x 1)(x 9) + (9 x 2 ) 10. 3(x 5)(x + 5) 11. -3(x 2)(x + 2) 12. x 2 20x 3x(x 10) POLY3 SP28

3.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 5 Fill in each missing expression and common factor. Expression as a sum GCF Expression as a product 1. 3x + 18 2. x 2 + 5x 3. 3x + 9y + 15 4. 4(5x + 8) 5. y(1 + y) 6. 10x 2 (x + 9) 7. 8 + 24y 8. 7x 2 + 14x 9. x 3 + 6x 2 + 7x 10. 4x 3 + 8x 2 + 12x 11. At 3:00 pm, Paul leaves Emoryville on a train traveling 45 mph. One hour later, Roger leaves Emoryville on a train traveling 60 mph. They arrive at Truckee at the same time. At what time do they arrive? Complete the table to determine when they arrive. Use algebra to confirm when they arrive. Time Car Train POLY3 SP29

3.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 6 Use what you know from observing patterns. Write all of the possible polynomials with integer coefficients for these sums and products. Use extra paper to sketch rectangles if needed. 1. polynomial as a sum of terms polynomial as a product of factors x 2 x + 12 (x ) (x ) 2. polynomial as a sum of terms polynomial as a product of factors x 2 + x + 16 (x + ) (x + ) Use what you know from observing patterns to help you factor each polynomial. 3. x 2 5x + 4 4. x 2 5x + 6 5. x 2 5x 6 6. 25 y 2 7. 3x 2 + 9x 8. x 2 100y 2 9. x 2 10x + 24 10. x 2 10x 24 11. x 2 + 10x 24 12. x 2 16 13. 12x 24x 2 14. 4x 2 100y 2 POLY3 SP30

3.4 Vocabulary, Skill Builders, and Review TEST PREPARATION (POLY3) Show your work on a separate sheet of paper and choose ALL correct answers. 1. Multiply (3x 1)(5x + 3) A. 15x 2 + 2x 3 B. 15x 2 + 2x + 3 C. 15x 2 4x 3 D. 15x 2 + 4x 3 2. The factors of x 2 4x 5 are A. (x 5) C. (x 1) 2. The factors of 64 x 2 are A. 16 + x B. (x + 1) D. (x + 5) A. 8 x B. 16 x C. 8 + x 3. Factor completely: 3x 2 15x + 12 A. (x 4)(3x 3) B. 3(x 2 5x + 4) C. 3(x 4)(x 1) 4. Factor completely: x 2 100 D. 3(x 2)(x 2) A. (x 100) 2 B. (x + 100) 2 C. (x 10)(x + 10) D. x(x 10) 5. If area of a rectangle is represented as x 2 + 14x + 24, then one of the dimensions is A. x + 2 C. x + 6 6. What is the greatest common factor of 9x 2 and 6x? B. x + 4 D. x + 12 A. 3 B. 3x C. 18x 2 D. 54x 3 POLY3 SP31

3.4 Vocabulary, Skill Builders, and Review KNOWLEDGE CHECK (POLY3) POLY 3.1 An Introduction to Factoring Polynomials 1. What are two patterns to look for when factoring a binomial? Factor each of the following. 2. 12x 2 30x 3. 64x 2 1 4. 128 8x 2 POLY 3.2 Exploring Factoring Patterns with a Model 5. Create three different rectangles using one x 2 piece, 16 unit pieces, and as many x-pieces as you wish. Write the polynomials represented by these rectangles as a sum of terms and as a product of factors. polynomial as sum of terms x 2 + x + 16 POLY 3.3 Factoring Trinomials Factor each trinomial completely. polynomial as product of factors (x + ) (x + ) 6. x 2 13x + 40 7. x 2 13x 30 8. 10x 2 + 19x + 6 POLY3 SP32

HOME-SCHOOL CONNECTION (POLY 3) Here are some questions to review topics from these lessons with your young mathematician. Factor each polynomial. 1. x 2 + 9x + 20 2. x 2 9x + 20 3. x 2 + x 20 4. x 2 x 20 5. Explain how you know that the signs of the binomial factors are correct in each of the factored solutions above. 6. Evelyn says that 6x 2 + 50x 100 = 2(3x 5)(x + 10). Is she correct? Show work with diagrams to explain. Signature Date POLY3 SP33

COMMON CORE STATE STANDARDS MATHEMATICS A-SSE-1a (CA.A.7a) A-SSE-2 CA Addition (CA.A.8a) CA Addition (CA.A.8b) A-APR-1 MP1 MP2 MP3 MP4 MP5 MP6 MP7 MP8 STANDARDS FOR MATHEMATICAL CONTENT Interpret expressions that represent a quantity in terms of its context: Interpret parts of an expression, such as terms, factors, and coefficients. Use the structure of an expression to identify ways to rewrite it. For example, see x 4 y 4 as (x 2 ) 2 (y 2 ) 2, thus recognizing it as a difference of squares that can be factored as (x 2 y 2 )(x 2 + y 2 ). Use the distributive property to express a sum of terms with a common factor as a multiple of a sum of terms with no common factor. For example, express xy 2 + x 2 y as xy(y + x). Use the properties of operations to express a product of a sum of terms as a sum of products. For example, use the properties of operations to express (x + 5)(3 - x + c) as -x 2 + cx - 2x + 5c + 15. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials STANDARDS FOR MATHEMATICAL PRACTICE Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. First Printing DO NOT DUPLICATE 2012 POLY3 SP34