1-3 Multiplying Polynomials. Find each product. 1. (x + 5)(x + 2)

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1 6. (a + 9)(5a 6) 1- Multiplying Polynomials Find each product. 1. (x + 5)(x + ) 7. FRAME Hugo is designing a frame as shown. The frame has a width of x inches all the way around. Write an expression that represents the total area of the picture and frame.. (y )(y + 4) The total length is x + 0 and the width is x (b 7)(b + ) Find each product. 4. (4n + )(n + 9) 8. (a 9)(a + 4a 4) 5. (8h 1)(h ) 9. (4y )(4y + 7y + ) 6. (a + 9)(5a 6) 7. FRAME Hugo is designing a frame as shown. The frame has a width of x inches all the way around. Write an expression that represents the total area of the picture and frame. 10. (x 4x + 5)(5x + x 4) Page 1

2 1. (g + 10)(g 5) 1- Multiplying Polynomials 10. (x 4x + 5)(5x + x 4) 14. (6a + 5)(5a + ) 15. (4x + 1)(6x + ) 16. (5y 4)(y 1) 11. (n + n 6)(5n n 8) 17. (6d 5)(4d 7) 18. (m + 5)(m + ) 19. (7n 6)(7n 6) Find each product. 1. (c 5)(c + ) 0. (1t 5)(1t + 5) 1. (g + 10)(g 5) 1. (5r + 7)(5r 7) 14. (6a + 5)(5a + ). (8w + 4x)(5w 6x) 15. (4x + 1)(6x + ) Page

3 1. (5r + 7)(5r 7) 1- Multiplying Polynomials. (8w + 4x)(5w 6x) 6. (4a + 7)(9a + a 7). (11z 5y)(z + y) 7. (m 5m + 4)(m + 7m ) 4. GARDEN A walkway surrounds a rectangular garden. The width of the garden is 8 feet, and the length is 6 feet. The width x of the walkway around the garden is the same on every side. Write an expression that represents the total area of the garden and walkway. Let x + 8 = the width of the garden and walkway and let x + 6 = the length of the garden and walkway. Find each product. 5. (y 11)(y y + ) 8. (x + 5x 1)(5x 6x + 1) 6. (4a + 7)(9a + a 7) 9. (b 4b 7)(b b 9) 7. (m 5m + 4)(m + 7m ) Page

4 1- Multiplying Polynomials CCSS STRUCTURE Find an expression to represent the area of each shaded region. 9. (b 4b 7)(b b 9). Find the area of the circle. 0. (6z 5z )(z z 4) Find the area of the rectangle. Simplify. 1. (m + )[(m + m 6) + (m m + 4)] Subtract the area of the rectangle from the area of the circle.. [(t + t 8) (t t + 6)](t 4) The area of the shaded region is represented by the expression 4πx + 1πx + 9π x 5x. CCSS STRUCTURE Find an expression to represent the area of each shaded region.. 4. Find the area of the rectangle. Page 4

5 The area of the shaded region is represented by the 1- Multiplying Polynomials expression 4πx + 1πx + 9π x 5x. The area of the shaded region is represented by the expression 4x. 5. VOLLEYBALL The dimensions of a sand volleyball court are represented by a width of 6y 5 feet and a length of y + 4 feet. a. Write an expression that represents the area of the court. 4. Find the area of the rectangle. b. The length of a sand volleyball court is 1 feet. Find the area of the court. a. Find the area of the triangle. The area of the court is represented by the expression 18y + 9y 0. b. Subtract the area of the triangle from the area of the rectangle. The area of the shaded region is represented by the expression 4x Substitute 9 for y in the expression for area to find the area of the sand volleyball court when the length is 1 feet. The area of the sand volleyball court is 1519 ft.. 5. VOLLEYBALL The dimensions of a sand volleyball court are represented by a width of 6y 5 feet and a length of y + 4 feet. 6. GEOMETRY Write an expression for the area of a triangle with a base of x + and a height of x 1. a. Write an expression that represents the area of the court. b. The length of a sand volleyball court is 1 feet. Find the area of the court. a. Page 5

6 1- Multiplying Polynomials The area of the sand volleyball court is 1519 ft. 6. GEOMETRY Write an expression for the area of a triangle with a base of x + and a height of x (r t) The area of the triangle is represented by the expression. Find each product. 7. (a b) 41. (5g + h) 8. (c + 4d) 4. (4y + z)(4y z) 9. (x 5y) 40. (r t) Page 6 4. CONSTRUCTION A sandbox kit allows you to build a square sandbox or a rectangular sandbox as

7 1- Multiplying Polynomials 4. (4y + z)(4y z) 4. CONSTRUCTION A sandbox kit allows you to build a square sandbox or a rectangular sandbox as shown. a. What are the possible values of x? Explain. 4. CONSTRUCTION A sandbox kit allows you to build a square sandbox or a rectangular sandbox as shown. b. Which shape has the greater area? c. What is the difference in areas between the two? a. The value of x must be greater than 4. If x = 4 the width of the rectangular sandbox would be zero and if x < 4 the width of the rectangular sandbox would be negative. b. a. What are the possible values of x? Explain. b. Which shape has the greater area? c. What is the difference in areas between the two? a. The value of x must be greater than 4. If x = 4 the width of the rectangular sandbox would be zero and if x < 4 the width of the rectangular sandbox would be negative. b. The square has the greatest area. c. Subtract the area of the rectangle from the area of the square. The difference in the areas is 4 ft. 44. MULTIPLE REPRESENTATIONS In this proble Page 7 a. TABULAR Copy and complete the table for ea

8 1- Multiplying Polynomials The difference in the areas is 4 ft. Then, 44. MULTIPLE REPRESENTATIONS In this proble a. TABULAR Copy and complete the table for ea 45. REASONING Determine if the following statement is sometimes, always, or never true. Explain your reasoning. The FOIL method can be used to multiply a binomial and a trinomial. Always; by grouping two adjacent terms, a trinomial can be written as a binomial (the sum of two quantities), and apply the FOIL method. For b. VERBAL Make a conjecture about the terms of c. SYMBOLIC For a sum of the form a + b, write a. example, (x + )( x + 5x + 7) = (x + )[ x + (5x + 7)] = x(x ) + x(5x + 7) + (x ) + (5x + 7). Then use the Distributive Property and simplify. m p 46. CHALLENGE Find (x + x )(x m 1 x 1 p p + x ). 47. OPEN ENDED Write a binomial and a trinomial involving a single variable. Then find their product. Sample answer: x 1, x x 1. b. The first term of the square of a sum is the first t times the first term of the sum multiplied by the last t last term of the sum squared. c. 48. CCSS REGULARITY Compare and contrast the procedure used to multiply a trinomial by a binomial using the vertical method with the procedure used to multiply a three-digit number by a two-digit number. Then, 45. REASONING Determine if the following statement is sometimes, always, or never true. Explain your reasoning. The FOIL method can be used to multiply a binomial and a trinomial. Always; by grouping two adjacent terms, a trinomial can be written as a binomial (the sum of two quantities), and apply the FOIL method. For example, (x + )( x + 5x + 7) = (x + )[ x + (5x The three monomials that make up the trinomial are similar to the three digits that make up the -digit number. The single monomial is similar to a 1-digit number. With each procedure you perform multiplications. The difference is that polynomial multiplication involves variables and the resulting product is often the sum of two or more monomials, while numerical multiplication results in a single number. Consider the following examples. Page 8

9 1- Multiplying Polynomials 48. CCSS REGULARITY Compare and contrast the procedure used to multiply a trinomial by a binomial using the vertical method with the procedure used to multiply a three-digit number by a two-digit number. The three monomials that make up the trinomial are similar to the three digits that make up the -digit number. The single monomial is similar to a 1-digit number. With each procedure you perform multiplications. The difference is that polynomial multiplication involves variables and the resulting product is often the sum of two or more monomials, while numerical multiplication results in a single number. 49. WRITING IN MATH Summarize the methods tha multiply polynomials. The Distributive Property can be used with a vertical distributing, multiplying, and combining like terms. Horizontal: Consider the following examples. The FOIL method is used with a horizontal format. Y outer, inner, and last terms of the binomials and then 49. WRITING IN MATH Summarize the methods tha multiply polynomials. A rectangular method can also be used by writing the polynomials along the top and left side of a rectangle terms and combining like terms. The Distributive Property can be used with a vertical distributing, multiplying, and combining like terms. Horizontal: 50. What is the product of x 5 and x + 4? A 5x 1 B 6x 7x 0 C 6x 0 D 6x + 7x 0 The FOIL method is used with a horizontal format. Y outer, inner, and last terms of the binomials and then Choice B is the correct answer. A rectangular method can also be used by writing the polynomials along the top and left side of a rectangle terms and combining like terms. 51. Which statement is correct about the symmetry of this design? Page 9

10 axis, you can eliminate this choice. J Since the figure is symmetrical about the y -axis, 1- Multiplying Polynomials Choice B is the correct answer. 51. Which statement is correct about the symmetry of this design? you can eliminate this choice. Thus, Choice F is the correct answer. 5. Which point on the number line represents a number that, when cubed, will result in a number greater than itself? A P B Q C R F The design is symmetrical only about the y-axis. D T G The design is symmetrical only about the x-axis. T is the only number greater than 1, so it is the only number when cubed that will be greater than itself. H The design is symmetrical about both the y- and the x-axes. Choice D is the correct answer. J The design has no symmetry. Consider each choice. 5. SHORT RESPONSE For a science project, Jodi selected three bean plants of equal height. Then, for five days, she measured their heights in centimeters and plotted the values on the graph below. F For the design to be symmetrical only about the yaxis, you can fold it along the y-axis. The part to the right and left of the y-axis should be identical. In this case they are. So the figure is symmetrical about the y-axis. G For the design to be symmetrical about the xaxis, you can fold it on the x-axis. The part above and below the x-axis, should be identical. In this case they are not. So it is not symmetrical about the xaxis. H Since the figure is not symmetrical about the x- axis, you can eliminate this choice. J Since the figure is symmetrical about the y -axis, you can eliminate this choice. She drew a line of best fit on the graph. What is the slope of the line that she drew? The line passes through the points (1, 1) and (5, 7). Thus, Choice F is the correct answer. 5. Which point on the number line represents a number that, when cubed, will result in a number greater than itself? Page 10

11 T is the only number greater than 1, so it is the only number when cubed that will be greater than itself. 1- Multiplying Polynomials Choice D is the correct answer. 5. SHORT RESPONSE For a science project, Jodi selected three bean plants of equal height. Then, for five days, she measured their heights in centimeters and plotted the values on the graph below. So, the slope of the line is. 54. SAVINGS Carrie has $6000 to invest. She puts x dollars of this money into a savings account that earns % interest per year. She uses the rest of the money to purchase a certificate of deposit that earns 4% interest. Write an equation for the amount of money that Carrie will have in one year. Let x = the amount placed into the % interest savings account Let 6000-x = the amount placed into the 4% certificate of deposit To calculate the amount of money that will be in the account at the end of the year, use principle (1 + rate) time. (The 1 + the rate will add back in the original money deposited.) She drew a line of best fit on the graph. What is the slope of the line that she drew? Savings account: The line passes through the points (1, 1) and (5, 7). Certificate of deposit: Therefore, T = 1.0x (6000 x) So, the slope of the line is. Find each sum or difference. 55. (7a 5) + ( a + 10) 54. SAVINGS Carrie has $6000 to invest. She puts x dollars of this money into a savings account that earns % interest per year. She uses the rest of the money to purchase a certificate of deposit that earns 4% interest. Write an equation for the amount of money that Carrie will have in one year. Let x = the amount placed into the % interest savings account Let 6000-x = the amount placed into the 4% certificate of deposit 56. (8n n ) + (4n 6n ) 57. (4 + n + n ) + (n 9n + 6) To calculate the amount of money that will be in the account at the end of the year, use principle (1 + rate) time. (The 1 + the rate will add back in the esolutions Manual - Powered by Cognero original money deposited.) Savings account: Page ( 4u 9 + u) + (6u u )

12 56. (8n n ) + (4n 6n ) 1- Multiplying Polynomials 57. (4 + n + n ) + (n 9n + 6) Simplify ( t ) ( t ) 58. ( 4u 9 + u) + (6u u ) 64. ( h ) ( h ) 59. (b + 4) + (c + b ) 65. ( 5y ) + ( y ) 60. (a 6a) (a + 5a) 61. ( 4m m + 10) (m + m 7) ( 6n ) + ( n ) 6. (a + 4ab + b) (b + 5a + 8ab) Simplify ( t ) ( t ) Page 1