# 8-5 Using the Distributive Property. Use the Distributive Property to factor each polynomial b 15a SOLUTION:

Size: px
Start display at page:

Download "8-5 Using the Distributive Property. Use the Distributive Property to factor each polynomial. 1. 21b 15a SOLUTION:" Transcription

1 Use the Distributive Property to factor each polynomial. 1. 1b 15a The greatest common factor in each term is c + c The greatest common factor in each term is c g h + 9gh g h The greatest common factor in each term is gh. 4. 1j k + 6j k + j k The greatest common factor in each term is j k. Factor each polynomial. 5. np + n + 8p + 16 Page 1

2 The greatest common factor in each term is j k. Factor each polynomial. 5. np + n + 8p xy 7x + 7y bc b c 8. 9fg 45f 7g + 35 Solve each equation. Check your solutions. 9. 3k(k + 10) = 0 Since 0 is on one side of the equation and the other side is in factor form, apply the Zero Product Property and set each factor equal to 0. Solve each of the resulting equations. 3k(k + 10) = 0 The roots are 0 and 10. Check by substituting 0 and 10 in for k in the original equation. So, the solutions are 0 and 10. Page

3 Solve each equation. Check your solutions. 9. 3k(k + 10) = 0 Since 0 is on one side of the equation and the other side is in factor form, apply the Zero Product Property and set each factor equal to 0. Solve each of the resulting equations. 3k(k + 10) = 0 The roots are 0 and 10. Check by substituting 0 and 10 in for k in the original equation. So, the solutions are 0 and (4m + )(3m 9) = 0 Since 0 is on one side of the equation and the other side is in factor form, apply the Zero Product Property and set each factor equal to 0. Solve each of the resulting equations. (4m + )(3m 9) = 0 The roots are and 3. Check by substituting and 3 in for m in the original equation. So, the solutions are and p 15p = 0 Since 0 is on one side of the equation, factor the other side. Next, apply the Zero Product Property by setting each factor equal to 0. Solve each of the resulting equations. esolutions Manual - Powered by Cognero Page 3

4 8-5 Using Distributive Property So, thethe solutions are and p 15p = 0 Since 0 is on one side of the equation, factor the other side. Next, apply the Zero Product Property by setting each factor equal to 0. Solve each of the resulting equations. The roots are 0 and. Check by substituting 0 and in for p in the original equation. So, the solutions are 0 and. 1. r = 14r Rewrite the equation so one side has 0 and the other side is in factor form. Next, apply the Zero Product Property by setting each factor equal to 0. Solve each of the resulting equations. r = 0 or The roots are 0 and 14. Check by substituting 0 and 14 in for r in the original equation. So, the solutions are 0 and SPIDERS Jumping spiders can commonly be found in homes and barns throughout the United States. A jumping spider s jump can be modeled by the equation h = 33.3t 16t, where t represents the time in seconds and h is the height in feet. a. When is the spider s height at 0 feet? b. What spider s height after 1 second? after seconds? esolutions Manualis- the Powered by Cognero Page 4 a. Write the equation so it is of the form ab = 0.

5 8-5 Using the Distributive Property So, the solutions are 0 and SPIDERS Jumping spiders can commonly be found in homes and barns throughout the United States. A jumping spider s jump can be modeled by the equation h = 33.3t 16t, where t represents the time in seconds and h is the height in feet. a. When is the spider s height at 0 feet? b. What is the spider s height after 1 second? after seconds? a. Write the equation so it is of the form ab = 0. t = 0 or The spider s height is at 0 ft. at 0 sec. and at.0815 sec. b. h = 33.3(1) 16(1) = 17.3 h = 33.3() 16() =.6 The spider s height after 1 second is 17.3 ft. and after seconds is.6 ft. 14. CCSS REASONING At a Fourth of July celebration, a rocket is launched straight up with an initial velocity of 15 feet per second. The height h of the rocket in feet above sea level is modeled by the formula h = 15t 16t, where t is the time in seconds after the rocket is launched. a. What is the height of the rocket when it returns to the ground? b. Let h = 0 in the equation and solve for t. c. How many seconds will it take for the rocket to return to the ground? a. The height of the rocket when it returns to the ground is 0 ft. b. t = 0 or t = 0 or c. It will take the rocket about 7.8 seconds to return to the ground. Use the Distributive Property to factor each polynomial t 40y The greatest common factor in each term is v + 50x esolutions Manual - Powered by Cognero Page 5

6 The greatest common factor in each term is v + 50x The greatest common factor in each term is k + 4k The greatest common factor in each term is k z + 10z The greatest common factor in each term is 5z a b + a b 10ab The greatest common factor in each term is ab c v 15c v + 5c v The greatest common factor in each term is 5c v. Page 6

7 The greatest common factor in each term is ab c v 15c v + 5c v The greatest common factor in each term is 5c v. Factor each polynomial. 1. fg 5g + 4f 0. a 4a 4 + 6a 3. hj h + 5j xy x + y Page 7

8 4. xy x + y 5. 45pq 7q 50p ty 18t + 4y dt 1d t 8. 8r + 1r 9. 1th 3t 35h + 5 Sometimes you need to reorder the terms so that there are common factors in each group. It is important that after nd step, there is a common factor. If not, go back and re-group the original terms and factor GCF's again. Page 8

9 9. 1th 3t 35h + 5 Sometimes you need to reorder the terms so that there are common factors in each group. It is important that after nd step, there is a common factor. If not, go back and re-group the original terms and factor GCF's again. 30. vp + 1v + 8p + 96 Sometimes you need to reorder the terms so that there are common factors in each group. It is important that after nd step, there is a common factor. If not, go back and re-group the original terms and factor GCF's again br 5b + r nu 8u + 3n 1 Sometimes you need to reorder the terms so that there are common factors in each group. It is important that after nd step, there is a common factor. If not, go back and re-group the original terms and factor GCF's again gf + g f + 15gf esolutions - Powered 34. rp Manual 9r + 9p 81 by Cognero Page 9 Sometimes you need to reorder the terms so that there are common factors in each group. It is important that after

10 33. 5gf + g f + 15gf 34. rp 9r + 9p 81 Sometimes you need to reorder the terms so that there are common factors in each group. It is important that after nd step, there is a common factor. If not, go back and re-group the original terms and factor GCF's again cd 18c d + 3cd r t + 1r t 6r t tu 90t + 3u gh + 4g h 3 Sometimes you need to reorder the terms so that there are common factors in each group. It is important that after nd step, there is a common factor. If not, go back and re-group the original terms and factor GCF's again. Solve each equation. Check your solutions b(9b 7) = 0 Page 10 Since 0 is on one side of the equation and the other side is in factor form, apply the Zero Product Property and set each factor equal to 0. Solve each of the resulting equations.

11 Solve each equation. Check your solutions b(9b 7) = 0 Since 0 is on one side of the equation and the other side is in factor form, apply the Zero Product Property and set each factor equal to 0. Solve each of the resulting equations. 3b(9b 7) = 0 The roots are 0 and 3. Check by substituting 0 and 3 in for b in the original equation. So, the solutions are 0 and n(3n + 3) = 0 Since 0 is on one side of the equation and the other side is in factor form, apply the Zero Product Property and set each factor equal to 0. Solve each of the resulting equations. n(3n + 3) = 0 The roots are 0 and 1. Check by substituting 0 and 1 in for n in the original equation. So, the solutions are 0 and (8z + 4)(5z + 10) = 0 Since 0 is on one side of the equation and the other side is in factor form, apply the Zero Product Property and set each factor equal to 0. Solve each of the resulting equations. (8z + 4)(5z + 10) = 0 The roots are and. Check by substituting Page 11 and in for z in the original equation.

12 So, the solutions are 0 and (8z + 4)(5z + 10) = 0 Since 0 is on one side of the equation and the other side is in factor form, apply the Zero Product Property and set each factor equal to 0. Solve each of the resulting equations. (8z + 4)(5z + 10) = 0 The roots are and. Check by substituting and in for z in the original equation. So, the solutions are and. 4. (7x + 3)(x 6) = 0 Since 0 is on one side of the equation and the other side is in factor form, apply the Zero Product Property and set each factor equal to 0. Solve each of the resulting equations. (7x + 3)(x 6) = 0 The roots are and 3. Check by substituting and 3 in for x in the original equation. So, the solutions are and b = 3b Page 1 Rewrite the equation so one side has 0 and the other side is in factor form. Next, apply the Zero Product Property by setting each factor equal to 0. Solve each of the resulting equations.

13 8-5 Using Distributive Property So, thethe solutions are and b = 3b Rewrite the equation so one side has 0 and the other side is in factor form. Next, apply the Zero Product Property by setting each factor equal to 0. Solve each of the resulting equations. b = 0 or The roots are 0 and 3. Check by substituting 0 and 3 in for b in the original equation. So, the solutions are 0 and a = 4a Rewrite the equation so one side has 0 and the other side is in factor form. Next, apply the Zero Product Property by setting each factor equal to 0. Solve each of the resulting equations. a = 0 or The roots are 0 and 4. Check by substituting 0 and 4 in for a in the original equation. So, the solutions are 0 and CCSS SENSE-MAKING Use the drawing shown. a. Write an expression in factored form to represent the area of the blue section. b. Write an expression in factored form to represent the area of the region formed by the outer edge. c. Write an expression in factored form to represent the yellow region. a. ab b. (a + 6)(b + 6) esolutions by Cognero c. (amanual + 6)(b- Powered + 6) ab = ab + 6a + 6b + 36 ab = 6a + 6b + 36 = 6(a + b + 6) Page 13

14 8-5 Using the Distributive Property So, the solutions are 0 and CCSS SENSE-MAKING Use the drawing shown. a. Write an expression in factored form to represent the area of the blue section. b. Write an expression in factored form to represent the area of the region formed by the outer edge. c. Write an expression in factored form to represent the yellow region. a. ab b. (a + 6)(b + 6) c. (a + 6)(b + 6) ab = ab + 6a + 6b + 36 ab = 6a + 6b + 36 = 6(a + b + 6) 46. FIREWORKS A ten-inch fireworks shell is fired from ground level. The height of the shell in feet is given by the formula h = 63t 16t, where t is the time in seconds after launch. a. Write the expression that represents the height in factored form. b. At what time will the height be 0? Is this answer practical? Explain. c. What is the height of the shell 8 seconds and 10 seconds after being fired? d. At 10 seconds, is the shell rising or falling? a. b. t = 0 or The answer is practical because the shell starts at ground level where t = 0 and h = 0 and is in the air for seconds before landing on the ground (h = 0) again. c. After 8 seconds the shell is 1080 ft. high. After 10 seconds, the shell is 1030 ft. high. d. Because the height is lower at 10 seconds than it is at 8 seconds, the shell has begun to fall. 47. ARCHITECTURE The frame of a doorway is an arch that can be modeled by the graph of the equation y = 3x + 1x, where x and y are measured in feet. On a coordinate plane, the floor is represented by the x-axis. a. Make a table of values for the height of the arch if x = 0, 1,, 3, and 4 feet. b. Plot the points from the table on a coordinate plane and connect the points to form a smooth curve to represent the arch. esolutions Manual - Powered Cognero Page 14 c. How high is the by doorway?

15 After 10 the shell is 1030 ft. high. 8-5 Using theseconds, Distributive Property d. Because the height is lower at 10 seconds than it is at 8 seconds, the shell has begun to fall. 47. ARCHITECTURE The frame of a doorway is an arch that can be modeled by the graph of the equation y = 3x + 1x, where x and y are measured in feet. On a coordinate plane, the floor is represented by the x-axis. a. Make a table of values for the height of the arch if x = 0, 1,, 3, and 4 feet. b. Plot the points from the table on a coordinate plane and connect the points to form a smooth curve to represent the arch. c. How high is the doorway? a. x y = 3x + 1x y 0 1 3(0) + 1(0) 3(1) + 1(1) 0 9 3() + 1() 1 3 3(3) + 1(3) 9 4 3(4) + 1(4) 0 b. c. The maximum point occurs at x = with a height of 1. Thus, the doorway is 1 ft high because 1 ft is the maximum height. 48. RIDES Suppose the height of a rider after being dropped can be modeled by h = 16t 96t + 160, where h is the height in feet and t is time in seconds. a. Write an expression to represent the height in factored form. b. From what height is the rider initially dropped? c. At what height will the rider be after 3 seconds of falling? Is this possible? Explain. a. 16t 96t = 16( t 6t + 10) b. The constant in the expression is 160, so the rider is initially dropped from 160 ft. c. No, this is not possible. The rider cannot be a negative number of feet in the air. 49. ARCHERY The height h in feet of an arrow can be modeled by the equation h = 64t 16t, where t is time in seconds. Ignoring the height of the archer, how long after it is released does it hit the ground? Page 15

16 No, this is not possible. The rider cannot be a negative number of feet in the air. 49. ARCHERY The height h in feet of an arrow can be modeled by the equation h = 64t 16t, where t is time in seconds. Ignoring the height of the archer, how long after it is released does it hit the ground? or It takes the arrow 4 seconds to hit the ground. 50. TENNIS A tennis player hits a tennis ball upward with an initial velocity of 80 feet per second. The height h in feet of the tennis ball can be modeled by the equation h = 80t 16t, where t is time in seconds. Ignoring the height of the tennis player, how long does it take the ball to hit the ground? It takes the ball 5 seconds to hit the ground. 51. MULTIPLE REPRESENTATIONS In this problem, you will explore the box method of factoring. To factor x + x 6, write the first term in the top left-hand corner of the box, and then write the last term in the lower right-hand corner. a. ANALYTICAL Determine which two factors have a product of 6 and a sum of 1. b. SYMBOLIC Write each factor in an empty square in the box. Include the positive or negative sign and variable. c. ANALYTICAL Find the factor for each row and column of the box. What are the factors of x + x 6? d. VERBAL Describe how you would use the box method to factor x 3x 40. a. Make a table with the different factors of 6. Find the sum and product of each pair of factors. Product Factor Factor Sum of of 1 Factors Factors esolutions by Cognero The Manual factors- Powered are 3 and since 3 + ( ) = 1 and 3( ) = 6. b. Page 16

17 Factors Using the Distributive Property The factors are 3 and since 3 + ( ) = 1 and 3( ) = 6. b. c. (x + 3)(x ) d. Place x in the top left-hand corner and place 40 in the lower right-hand corner. Then determine which two factors have a product of 40 and a sum of 3. Then place these factors in the box. Then find the factor of each row and column. The factors will be listed on the very top and far left of the box. 5. CCSS CRITIQUE Hernando and Rachel are solving m = 4m. Is either of them correct? Explain your reasoning. Page 17 Hernando divided one side by m and the other by m. The Division Property of Equality states that each side must be divided by the same quantity. Also, since m is a variable it could equal 0. Division by 0 does not produce an equivalent equation. Rachel has applied the Zero Product Property by rewriting the equation so that one side has

18 5. CCSS CRITIQUE Hernando and Rachel are solving m = 4m. Is either of them correct? Explain your reasoning. Hernando divided one side by m and the other by m. The Division Property of Equality states that each side must be divided by the same quantity. Also, since m is a variable it could equal 0. Division by 0 does not produce an equivalent equation. Rachel has applied the Zero Product Property by rewriting the equation so that one side has zero and the other side is in factor form. Then she set each factor equal to 0 and solved. Rachel is correct. 53. CHALLENGE Given the equation (ax + b)(ax b) = 0, solve for x. What do we know about the values of a and b? or If a = 0 and b = 0, then all real numbers are solutions. If a 0, then the solutions are and. 54. OPEN ENDED Write a four-term polynomial that can be factored by grouping. Then factor the polynomial. Write 4 monomials where there is a GCF for each group. Once the GCF is factored out, there should be another common factor. 55. REASONING Given the equation c = a ab, for what values of a and b does c = 0? Let c = 0, factor and solve. esolutions Manual - Powered by Cognero a ab = 0 a(a b) = 0 a = 0 or a = b Page 18

19 common factor. 55. REASONING Given the equation c = a ab, for what values of a and b does c = 0? Let c = 0, factor and solve. a ab = 0 a(a b) = 0 a = 0 or a = b Then c = 0 when a = 0 or a = b for any real values of a and b. 56. WRITING IN MATH Explain how to solve a quadratic equation by using the Zero Product Property. Rewrite the equation to have zero on one side of the equals sign. Then factor the other side. Set each factor equal to zero, and then solve each equation. For example solve x 5x = Which is a factor of 6z 3z + 4z? A z + 1 B 3z C z + D z 1 The correct choice is D. 58. PROBABILITY Hailey has 10 blocks: red, 4 blue, 3 yellow, and 1 green. If Hailey chooses one block, what is the probability that it will be either red or yellow? F G H J The total number of blocks is 10 and the number of red or yellow blocks is + 3, or 5. So the probability of choosing either a red or a yellow block is. Page 19 The correct choice is H. 59. GRIDDED RESPONSE Cho is making a 140-inch by 160-inch quilt with quilt squares that measure 8 inches on

20 The correct choice is D. 58. PROBABILITY Hailey has 10 blocks: red, 4 blue, 3 yellow, and 1 green. If Hailey chooses one block, what is the probability that it will be either red or yellow? F G H J The total number of blocks is 10 and the number of red or yellow blocks is + 3, or 5. So the probability of choosing either a red or a yellow block is. The correct choice is H. 59. GRIDDED RESPONSE Cho is making a 140-inch by 160-inch quilt with quilt squares that measure 8 inches on each side. How many will be needed to make the quilt? The area of the quilt is , or,400. The area of each square is 8 8, or 64. To find the number of squares needed, divide the area of the quilt by the area of each square:, = 350. So, 350 squares will be needed to make the quilt. 60. GEOMETRY The area of the right triangle shown is 5h square centimeters. What is the height of the triangle? A cm B 5 cm C 8 cm D 10 cm h=5 The height of the triangle is h, or (5) = 10 cm. The correct choice is D. 61. GENETICS Brown genes B are dominant over blue genes b. A person with genes BB or Bb has brown eyes. eyes. Elisa has brown eyes with Bb genes, and Bob has blue eyes. Write an Page 0 expression for the possible eye coloring of Elisa and Bob s children. Determine the probability that their child would have blue eyes. esolutions Manualwith - Powered Cognero Someone genesbybb has blue

21 h=5 The height of the triangleproperty is h, or (5) = 10 cm. 8-5 Using the Distributive The correct choice is D. 61. GENETICS Brown genes B are dominant over blue genes b. A person with genes BB or Bb has brown eyes. Someone with genes bb has blue eyes. Elisa has brown eyes with Bb genes, and Bob has blue eyes. Write an expression for the possible eye coloring of Elisa and Bob s children. Determine the probability that their child would have blue eyes. Bob has blue eyes so he must have bb genes. Elisa has brown eyes so she could have BB, Bb, or bb genes. Their children could get only a b gene from Bob but either a B or b gene from Elisa. The following table shows the combinations if each child would receive half of each parent s genes. So, an expression for the possible eye coloring of Elisa and Bob s children is: 0.5Bb + 0.5b. If B = 0 and b = 1, then Therefore, the probability that a child of Elisa and Bob would have blue eyes is. Find each product. 6. n(n 4n + 3) 63. b(b + b 5) 64. c(4c + c ) Page 1

22 64. c(4c + c ) x(x + x + x 1) 66. ab(4a b + ab b ) 67. 3xy(x + xy + y ) Simplify. esolutions Manual - Powered by Cognero (ab )(ab ) Page

23 Simplify (ab )(ab ) (p r )(p r) ( 7c d )(4cd ) (9xy ) BASKETBALL In basketball, a free throw is 1 point, and a field goal is either or 3 points. In a season, Tim Duncan of the San Antonio Spurs scored a total of 134 points. The total number of -point field goals and 3-point field goals was 517, and he made 305 of the 455 free throws that he attempted. Find the number of -point field goals and 3-point field goals Duncan made that season. Page 3 Let t = the number of 3-point field goals Duncan made. His total number of points can be found by the sum of the

24 74. BASKETBALL In basketball, a free throw is 1 point, and a field goal is either or 3 points. In a season, Tim Duncan of the San Antonio Spurs scored a total of 134 points. The total number of -point field goals and 3-point field goals was 517, and he made 305 of the 455 free throws that he attempted. Find the number of -point field goals and 3-point field goals Duncan made that season. Let t = the number of 3-point field goals Duncan made. His total number of points can be found by the sum of the points earned by his 3-point field goals, his -point field goals, and his free throws = 514 Duncan made three 3-point field goals and 514 -point field goals that season. Solve each inequality. Check your solution y 4 > 37 The solution is {y y > 11}. Check the solution by substituting a number greater than 11 for y in the equation. The solution checks q + 9 > 4 The solution is {q q < 3}. Check the solution by substituting a number less than 3 for q in the equation. The solution checks. + 1 < k Manual esolutions Powered by Cognero Page 4

25 The solution checks. 77. k + 1 < 30 The solution is {k k > 9}. Check the solution by substituting a number greater than 9 for k in the equation. The solution checks q + 7 3(q + 1) The solution is {q q }. Check the solution by substituting a number less than or equal to for q in the equation. The solution checks The solution is {z z 48}. Page 5

26 The solution checks The solution is {z z 48}. Check the solution by substituting a number greater than or equal to 48 for z in the equation. The solution checks c (c 5) > c + 17 The solution is {c c > }. Check the solution by substituting a number greater than for c in the equation. The solution checks. Find each product. 81. (a + )(a + 5) 8. (d + 4)(d + 10) Page 6

27 8. (d + 4)(d + 10) 83. (z 1)(z 8) 84. (c + 9)(c 3) 85. (x 7)(x 6) 86. (g )(g + 11) Page 7

### 5-3 Polynomial Functions. not in one variable because there are two variables, x. and y y. 5-3 Polynomial Functions State the degree and leading coefficient of each polynomial in one variable. If it is not a polynomial in one variable, explain why. 1. 11x 6 5x 5 + 4x 2 coefficient of the

### Factoring Polynomials UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can

### 15.1 Factoring Polynomials LESSON 15.1 Factoring Polynomials Use the structure of an expression to identify ways to rewrite it. Also A.SSE.3? ESSENTIAL QUESTION How can you use the greatest common factor to factor polynomials? EXPLORE

### Veterans Upward Bound Algebra I Concepts - Honors Veterans Upward Bound Algebra I Concepts - Honors Brenda Meery Kaitlyn Spong Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) www.ck12.org Chapter 6. Factoring CHAPTER

### Sect 6.7 - Solving Equations Using the Zero Product Rule Sect 6.7 - Solving Equations Using the Zero Product Rule 116 Concept #1: Definition of a Quadratic Equation A quadratic equation is an equation that can be written in the form ax 2 + bx + c = 0 (referred

### Factor Polynomials Completely 9.8 Factor Polynomials Completely Before You factored polynomials. Now You will factor polynomials completely. Why? So you can model the height of a projectile, as in Ex. 71. Key Vocabulary factor by grouping

### Factoring and Applications Factoring and Applications What is a factor? The Greatest Common Factor (GCF) To factor a number means to write it as a product (multiplication). Therefore, in the problem 48 3, 4 and 8 are called the

### Definitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder). Math 50, Chapter 8 (Page 1 of 20) 8.1 Common Factors Definitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder). Find all the factors of a. 44 b. 32

### How To Factor By Gcf In Algebra 1.5 7-2 Factoring by GCF Warm Up Lesson Presentation Lesson Quiz Algebra 1 Warm Up Simplify. 1. 2(w + 1) 2. 3x(x 2 4) 2w + 2 3x 3 12x Find the GCF of each pair of monomials. 3. 4h 2 and 6h 2h 4. 13p and 26p

### 10 7, 8. 2. 6x + 30x + 36 SOLUTION: 8-9 Perfect Squares. The first term is not a perfect square. So, 6x + 30x + 36 is not a perfect square trinomial. Squares Determine whether each trinomial is a perfect square trinomial. Write yes or no. If so, factor it. 1.5x + 60x + 36 SOLUTION: The first term is a perfect square. 5x = (5x) The last term is a perfect

### Introduction Assignment PRE-CALCULUS 11 Introduction Assignment Welcome to PREC 11! This assignment will help you review some topics from a previous math course and introduce you to some of the topics that you ll be studying

### Polynomials. Key Terms. quadratic equation parabola conjugates trinomial. polynomial coefficient degree monomial binomial GCF Polynomials 5 5.1 Addition and Subtraction of Polynomials and Polynomial Functions 5.2 Multiplication of Polynomials 5.3 Division of Polynomials Problem Recognition Exercises Operations on Polynomials

### 8-8 Differences of Squares. Factor each polynomial. 1. x 9 SOLUTION: 2. 4a 25 SOLUTION: 3. 9m 144 SOLUTION: 4. 2p 162p SOLUTION: 5. Factor each polynomial. 1.x 9 SOLUTION:.a 5 SOLUTION:.9m 1 SOLUTION:.p 16p SOLUTION: 5.u 81 SOLUTION: Page 1 5.u 81 SOLUTION: 6.d f SOLUTION: 7.0r 5n SOLUTION: 8.56n c SOLUTION: Page 8.56n c SOLUTION: 476 CHAPTER 7 Graphs, Equations, and Inequalities 7. Quadratic Equations Now Work the Are You Prepared? problems on page 48. OBJECTIVES 1 Solve Quadratic Equations by Factoring (p. 476) Solve Quadratic

### Polynomial Equations and Factoring 7 Polynomial Equations and Factoring 7.1 Adding and Subtracting Polynomials 7.2 Multiplying Polynomials 7.3 Special Products of Polynomials 7.4 Dividing Polynomials 7.5 Solving Polynomial Equations in

### In the above, the number 19 is an example of a number because its only positive factors are one and itself. Math 100 Greatest Common Factor and Factoring by Grouping (Review) Factoring Definition: A factor is a number, variable, monomial, or polynomial which is multiplied by another number, variable, monomial,

### Algebra I. In this technological age, mathematics is more important than ever. When students In this technological age, mathematics is more important than ever. When students leave school, they are more and more likely to use mathematics in their work and everyday lives operating computer equipment,

### MATH 10034 Fundamental Mathematics IV MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials. Introduction to Quadratic Functions The St. Louis Gateway Arch was constructed from 1963 to 1965. It cost 13 million dollars to build..1 Up and Down or Down and Up Exploring Quadratic Functions...617.2

### 7-2 Solving Exponential Equations and Inequalities. Solve each equation. 1. 3 5x = 27 2x 4 SOLUTION: 7-2 Solving Exponential Equations and Inequalities Solve each equation. 1. 3 5x = 27 2x 4 3. 2 6x = 32 x 2 12 2. 16 2y 3 = 4 y + 1 10 4. 49 x + 5 = 7 8x 6 3. 2 6x = 32 x 2 5. SCIENCE Mitosis is a process

### 1.1 Practice Worksheet Math 1 MPS Instructor: Cheryl Jaeger Balm 1 1.1 Practice Worksheet 1. Write each English phrase as a mathematical expression. (a) Three less than twice a number (b) Four more than half of a number (c)

### Factoring a Difference of Two Squares. Factoring a Difference of Two Squares 284 (6 8) Chapter 6 Factoring 87. Tomato soup. The amount of metal S (in square inches) that it takes to make a can for tomato soup is a function of the radius r and height h: S 2 r 2 2 rh a) Rewrite this

### Section 6.1 Factoring Expressions Section 6.1 Factoring Expressions The first method we will discuss, in solving polynomial equations, is the method of FACTORING. Before we jump into this process, you need to have some concept of what

### SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS (Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic

### BEGINNING ALGEBRA ACKNOWLEDMENTS BEGINNING ALGEBRA The Nursing Department of Labouré College requested the Department of Academic Planning and Support Services to help with mathematics preparatory materials for its Bachelor of Science

### How To Solve Factoring Problems 05-W4801-AM1.qxd 8/19/08 8:45 PM Page 241 Factoring, Solving Equations, and Problem Solving 5 5.1 Factoring by Using the Distributive Property 5.2 Factoring the Difference of Two Squares 5.3 Factoring

### Factoring (pp. 1 of 4) Factoring (pp. 1 of 4) Algebra Review Try these items from middle school math. A) What numbers are the factors of 4? B) Write down the prime factorization of 7. C) 6 Simplify 48 using the greatest common

### MATH 100 PRACTICE FINAL EXAM MATH 100 PRACTICE FINAL EXAM Lecture Version Name: ID Number: Instructor: Section: Do not open this booklet until told to do so! On the separate answer sheet, fill in your name and identification number

### 1.3 Algebraic Expressions 1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,

### Sample Problems. Practice Problems Lecture Notes Quadratic Word Problems page 1 Sample Problems 1. The sum of two numbers is 31, their di erence is 41. Find these numbers.. The product of two numbers is 640. Their di erence is 1. Find these 9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation

### Factoring Guidelines. Greatest Common Factor Two Terms Three Terms Four Terms. 2008 Shirley Radai Factoring Guidelines Greatest Common Factor Two Terms Three Terms Four Terms 008 Shirley Radai Greatest Common Factor 008 Shirley Radai Factoring by Finding the Greatest Common Factor Always check for

### NSM100 Introduction to Algebra Chapter 5 Notes Factoring Section 5.1 Greatest Common Factor (GCF) and Factoring by Grouping Greatest Common Factor for a polynomial is the largest monomial that divides (is a factor of) each term of the polynomial. GCF is the

### Factoring. 472 Chapter 9 Factoring Factoring Lesson 9- Find the prime factorizations of integers and monomials. Lesson 9- Find the greatest common factors (GCF) for sets of integers and monomials. Lessons 9-2 through 9-6 Factor polynomials.

### Summary of important mathematical operations and formulas (from first tutorial): EXCEL Intermediate Tutorial Summary of important mathematical operations and formulas (from first tutorial): Operation Key Addition + Subtraction - Multiplication * Division / Exponential ^ To enter a

### Algebra 1 If you are okay with that placement then you have no further action to take Algebra 1 Portion of the Math Placement Test Dear Parents, Based on the results of the High School Placement Test (HSPT), your child should forecast to take Algebra 1 this fall. If you are okay with that placement then you have no further action

### 6.1 Add & Subtract Polynomial Expression & Functions 6.1 Add & Subtract Polynomial Expression & Functions Objectives 1. Know the meaning of the words term, monomial, binomial, trinomial, polynomial, degree, coefficient, like terms, polynomial funciton, quardrtic

### Factor and Solve Polynomial Equations. In Chapter 4, you learned how to factor the following types of quadratic expressions. 5.4 Factor and Solve Polynomial Equations Before You factored and solved quadratic equations. Now You will factor and solve other polynomial equations. Why? So you can find dimensions of archaeological

### Polynomial Degree and Finite Differences CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial

### Factoring. Key Vocabulary 8 Factoring Find the prime factorization of integers and monomials. Factor polynomials. Use the Zero Product Property to solve equations. Key Vocabulary factored form (p. 41) perfect square trinomials

### Solutions to old Exam 1 problems Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections

### MATH 90 CHAPTER 6 Name:. MATH 90 CHAPTER 6 Name:. 6.1 GCF and Factoring by Groups Need To Know Definitions How to factor by GCF How to factor by groups The Greatest Common Factor Factoring means to write a number as product. a

### Lesson 1: Multiplying and Factoring Polynomial Expressions Lesson 1: Multiplying and Factoring Polynomial Expressions Student Outcomes Students use the distributive property to multiply a monomial by a polynomial and understand that factoring reverses the multiplication

### 6706_PM10SB_C4_CO_pp192-193.qxd 5/8/09 9:53 AM Page 192 192 NEL 92 NEL Chapter 4 Factoring Algebraic Epressions GOALS You will be able to Determine the greatest common factor in an algebraic epression and use it to write the epression as a product Recognize different

### Factoring Polynomials Factoring Polynomials 8A Factoring Methods 8-1 Factors and Greatest Common Factors Lab Model Factoring 8-2 Factoring by GCF Lab Model Factorization of Trinomials 8-3 Factoring x 2 + bx + c 8-4 Factoring

### 12-1 Representations of Three-Dimensional Figures Connect the dots on the isometric dot paper to represent the edges of the solid. Shade the tops of 12-1 Representations of Three-Dimensional Figures Use isometric dot paper to sketch each prism. 1. triangular

### Algebra I Vocabulary Cards Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression

### ESSENTIAL QUESTION How can you factor expressions of the form ax 2 + bx + c? LESSON 15.3 Factoring ax 2 + bx + c A.SSE.2 Use the structure of an expression to identify ways to rewrite it. Also A.SSE.3? ESSENTIAL QUESTION How can you factor expressions of the form ax 2 + bx + c?

### 1.3 Polynomials and Factoring 1.3 Polynomials and Factoring Polynomials Constant: a number, such as 5 or 27 Variable: a letter or symbol that represents a value. Term: a constant, variable, or the product or a constant and variable.

### 1-3 Distance and Midpoints. Use the number line to find each measure. and Midpoints The distance between W and Zis9So,WZ = 9 Use the number line to find each measure 1XY SOLUTION: TIME CAPSULE Graduating classes have buried time capsules on the campus of East Side High School

### Mathematics as Problem Solving The students will demonstrate the ability to gather information from a graphical representation of an equation. Title: Another Way of Factoring Brief Overview: Students will find factors for quadratic equations with a leading coefficient of one. The students will then graph these equations using a graphing calculator

### 2.3 Maximum and Minimum Applications Section.3 155.3 Maximum and Minimum Applications Maximizing (or minimizing) is an important technique used in various fields of study. In business, it is important to know how to find the maximum profit

### 5.1 FACTORING OUT COMMON FACTORS C H A P T E R 5 Factoring he sport of skydiving was born in the 1930s soon after the military began using parachutes as a means of deploying troops. T Today, skydiving is a popular sport around the world.

### 6.4 Factoring Polynomials Name Class Date 6.4 Factoring Polynomials Essential Question: What are some ways to factor a polynomial, and how is factoring useful? Resource Locker Explore Analyzing a Visual Model for Polynomial Factorization

### Quick Reference ebook This file is distributed FREE OF CHARGE by the publisher Quick Reference Handbooks and the author. Quick Reference ebook Click on Contents or Index in the left panel to locate a topic. The math facts listed

### MATH 60 NOTEBOOK CERTIFICATIONS MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5

### Algebra Cheat Sheets Sheets Algebra Cheat Sheets provide you with a tool for teaching your students note-taking, problem-solving, and organizational skills in the context of algebra lessons. These sheets teach the concepts

### Mathematics Placement Mathematics Placement The ACT COMPASS math test is a self-adaptive test, which potentially tests students within four different levels of math including pre-algebra, algebra, college algebra, and trigonometry.

### Wentzville School District Algebra 1: Unit 8 Stage 1 Desired Results Wentzville School District Algebra 1: Unit 8 Stage 1 Desired Results Unit Title: Quadratic Expressions & Equations Course: Algebra I Unit 8 - Quadratic Expressions & Equations Brief Summary of Unit: At

### 12) 13) 14) (5x)2/3. 16) x5/8 x3/8. 19) (r1/7 s1/7) 2 DMA 080 WORKSHEET # (8.-8.2) Name Find the square root. Assume that all variables represent positive real numbers. ) 6 2) 8 / 2) 9x8 ) -00 ) 8 27 2/ Use a calculator to approximate the square root to decimal

### Review of Intermediate Algebra Content Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6

### POLYNOMIALS and FACTORING POLYNOMIALS and FACTORING Exponents ( days); 1. Evaluate exponential expressions. Use the product rule for exponents, 1. How do you remember the rules for exponents?. How do you decide which rule to use

### FACTORING OUT COMMON FACTORS 278 (6 2) Chapter 6 Factoring 6.1 FACTORING OUT COMMON FACTORS In this section Prime Factorization of Integers Greatest Common Factor Finding the Greatest Common Factor for Monomials Factoring Out the

### Warm-Up Oct. 22. Daily Agenda: Evaluate y = 2x 3x + 5 when x = 1, 0, and 2. Daily Agenda: Grade Assignment Go over Ch 3 Test; Retakes must be done by next Tuesday 5.1 notes / assignment Graphing Quadratic Functions 5.2 notes / assignment

### Common Core Standards Practice Week 8 Common Core Standards Practice Week 8 Selected Response 1. Describe the end behavior of the polynomial f(x) 5 x 8 8x 1 6x. A down and down B down and up C up and down D up and up Constructed Response.

### Summer Math Exercises. For students who are entering. Pre-Calculus Summer Math Eercises For students who are entering Pre-Calculus It has been discovered that idle students lose learning over the summer months. To help you succeed net fall and perhaps to help you learn

### Name Intro to Algebra 2. Unit 1: Polynomials and Factoring Name Intro to Algebra 2 Unit 1: Polynomials and Factoring Date Page Topic Homework 9/3 2 Polynomial Vocabulary No Homework 9/4 x In Class assignment None 9/5 3 Adding and Subtracting Polynomials Pg. 332

### Factoring Polynomials Factoring Polynomials 8A Factoring Methods 8-1 Factors and Greatest Common Factors Lab Model Factorization by GCF 8-2 Factoring by GCF Lab Model Factorization of x 2 + bx + c 8-3 Factoring x 2 + bx + c

### SPECIAL PRODUCTS AND FACTORS CHAPTER 442 11 CHAPTER TABLE OF CONTENTS 11-1 Factors and Factoring 11-2 Common Monomial Factors 11-3 The Square of a Monomial 11-4 Multiplying the Sum and the Difference of Two Terms 11-5 Factoring the

### FACTORING POLYNOMIALS 296 (5-40) Chapter 5 Exponents and Polynomials where a 2 is the area of the square base, b 2 is the area of the square top, and H is the distance from the base to the top. Find the volume of a truncated

### 1) (-3) + (-6) = 2) (2) + (-5) = 3) (-7) + (-1) = 4) (-3) - (-6) = 5) (+2) - (+5) = 6) (-7) - (-4) = 7) (5)(-4) = 8) (-3)(-6) = 9) (-1)(2) = Extra Practice for Lesson Add or subtract. ) (-3) + (-6) = 2) (2) + (-5) = 3) (-7) + (-) = 4) (-3) - (-6) = 5) (+2) - (+5) = 6) (-7) - (-4) = Multiply. 7) (5)(-4) = 8) (-3)(-6) = 9) (-)(2) = Division is

### ALGEBRA I (Created 2014) Amherst County Public Schools ALGEBRA I (Created 2014) Amherst County Public Schools The 2009 Mathematics Standards of Learning Curriculum Framework is a companion document to the 2009 Mathematics Standards of Learning and amplifies

### 6.3 FACTORING ax 2 bx c WITH a 1 290 (6 14) Chapter 6 Factoring e) What is the approximate maximum revenue? f) Use the accompanying graph to estimate the price at which the revenue is zero. y Revenue (thousands of dollars) 300 200 100

### 6.6 Factoring Strategy 456 CHAPTER 6. FACTORING 6.6 Factoring Strategy When you are concentrating on factoring problems of a single type, after doing a few you tend to get into a rhythm, and the remainder of the exercises, because

### MATH 21. College Algebra 1 Lecture Notes MATH 21 College Algebra 1 Lecture Notes MATH 21 3.6 Factoring Review College Algebra 1 Factoring and Foiling 1. (a + b) 2 = a 2 + 2ab + b 2. 2. (a b) 2 = a 2 2ab + b 2. 3. (a + b)(a b) = a 2 b 2. 4. (a

### Factoring. 472 Chapter 9 Factoring Factoring Lesson 9- Find the prime factorizations of integers and monomials. Lesson 9- Find the greatest common factors (GCF) for sets of integers and monomials. Lessons 9-2 through 9-6 Factor polynomials.

### In this section, you will develop a method to change a quadratic equation written as a sum into its product form (also called its factored form). CHAPTER 8 In Chapter 4, you used a web to organize the connections you found between each of the different representations of lines. These connections enabled you to use any representation (such as a graph,

### Algebra 2 PreAP. Name Period Algebra 2 PreAP Name Period IMPORTANT INSTRUCTIONS FOR STUDENTS!!! We understand that students come to Algebra II with different strengths and needs. For this reason, students have options for completing

### 2.1. Inductive Reasoning EXAMPLE A CONDENSED LESSON 2.1 Inductive Reasoning In this lesson you will Learn how inductive reasoning is used in science and mathematics Use inductive reasoning to make conjectures about sequences of numbers

### 1.3. Maximum or Minimum of a Quadratic Function. Investigate A < P1-6 photo of a large arched bridge, similar to the one on page 292 or p 360-361of the fish book> Maximum or Minimum of a Quadratic Function 1.3 Some bridge arches are defined by quadratic functions.

### Polynomials and Factoring; More on Probability Polynomials and Factoring; More on Probability Melissa Kramer, (MelissaK) Anne Gloag, (AnneG) Andrew Gloag, (AndrewG) Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required)

### 1 of 7 9/5/2009 6:12 PM 1 of 7 9/5/2009 6:12 PM Chapter 2 Homework Due: 9:00am on Tuesday, September 8, 2009 Note: To understand how points are awarded, read your instructor's Grading Policy. [Return to Standard Assignment View]

### 2After completing this chapter you should be able to After completing this chapter you should be able to solve problems involving motion in a straight line with constant acceleration model an object moving vertically under gravity understand distance time

### ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola

### AIP Factoring Practice/Help The following pages include many problems to practice factoring skills. There are also several activities with examples to help you with factoring if you feel like you are not proficient with it. There

### What are the place values to the left of the decimal point and their associated powers of ten? The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything

### AP CALCULUS AB 2006 SCORING GUIDELINES. Question 4 AP CALCULUS AB 2006 SCORING GUIDELINES Question 4 t (seconds) vt () (feet per second) 0 10 20 30 40 50 60 70 80 5 14 22 29 35 40 44 47 49 Rocket A has positive velocity vt () after being launched upward

### 2010 Solutions. a + b. a + b 1. (a + b)2 + (b a) 2. (b2 + a 2 ) 2 (a 2 b 2 ) 2 00 Problem If a and b are nonzero real numbers such that a b, compute the value of the expression ( ) ( b a + a a + b b b a + b a ) ( + ) a b b a + b a +. b a a b Answer: 8. Solution: Let s simplify the

### Algebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year. This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra

### Factoring Polynomials and Solving Quadratic Equations Factoring Polynomials and Solving Quadratic Equations Math Tutorial Lab Special Topic Factoring Factoring Binomials Remember that a binomial is just a polynomial with two terms. Some examples include 2x+3

### FREE FALL. Introduction. Reference Young and Freedman, University Physics, 12 th Edition: Chapter 2, section 2.5 Physics 161 FREE FALL Introduction This experiment is designed to study the motion of an object that is accelerated by the force of gravity. It also serves as an introduction to the data analysis capabilities

### Applications for Triangles Not drawn to scale Applications for Triangles 1. 36 in. 40 in. 33 in. 1188 in. 2 69 in. 2 138 in. 2 1440 in. 2 2. 188 in. 2 278 in. 2 322 in. 2 none of these Find the area of a parallelogram with the given

### Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given. Polynomials (Ch.1) Study Guide by BS, JL, AZ, CC, SH, HL Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given. Sasha s method

### Algebra I Module 1 Lessons 1 28 Eureka Math 2015 2016 Algebra I Module 1 Lessons 1 28 Eureka Math, Published by the non-profit Great Minds. Copyright 2015 Great Minds. No part of this work may be reproduced, distributed, modified, sold,

### expression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method. A polynomial of degree n (in one variable, with real coefficients) is an expression of the form: a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 2 x 2 + a 1 x + a 0 where a n, a n 1, a n 2, a 2, a 1, a 0 are

### Factors and Products CHAPTER 3 Factors and Products What You ll Learn use different strategies to find factors and multiples of whole numbers identify prime factors and write the prime factorization of a number find square Algebra 1 End-of-Course Exam Practice Test with Solutions For Multiple Choice Items, circle the correct response. For Fill-in Response Items, write your answer in the box provided, placing one digit in We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical