Factoring. 472 Chapter 9 Factoring


 Marlene Cook
 5 years ago
 Views:
Transcription
1 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 92 through 96 Factor polynomials. Lessons 92 through 96 Use the Zero Product Property to solve equations. Key Vocabulary factored form (p. 475) factoring by grouping (p. 482) prime polynomial (p. 497) difference of squares (p. 50) perfect square trinomials (p. 508) The factoring of polynomials can be used to solve a variety of realworld problems and lays the foundation for the further study of polynomial equations. Factoring is used to solve problems involving vertical motion. For eample, the height h in feet of a dolphin that jumps out of the water traveling at 20 feet per second is modeled by a polynomial equation. Factoring can be used to determine how long the dolphin is in the air. You will learn how to solve polynomial equations in Lesson Chapter 9 Factoring
2 Prerequisite Skills To be successful in this chapter, you ll need to master these skills and be able to apply them in problemsolving situations. Review these skills before beginning Chapter 9. For Lessons 92 through 96 Distributive Property Rewrite each epression using the Distributive Property. Then simplify. (For review, see Lesson 5.). 3(4 ) 2. a(a 5) 3. 7(n 2 3n ) 4. 6y( 3y 5y 2 y 3 ) For Lessons 93 and 94 Multiplying Binomials Find each product. (For review, see Lesson 87.) 5. ( 4)( 7) 6. (3n 4)(n 5) 7. (6a 2b)(9a b) 8. ( 8y)(2 2y) For Lessons 95 and 96 Special Products Find each product. (For review, see Lesson 88.) 9. (y 9) 2 0. (3a 2) 2. (n 5)(n 5) 2. (6p 7q)(6p 7q) For Lesson 96 Square Roots Find each square root. (For review, see Lesson 27.) Make this Foldable to help you organize your notes on factoring. Begin with a sheet of plain 8 2 " by " paper. Fold in Siths Fold in thirds and then in half along the width. Fold Again Open. Fold lengthwise, leaving a " 2 tab on the right. Cut Label Open. Cut short side along folds to make tabs. Label each tab as shown F a ct o ri n g 96 Reading and Writing As you read and study the chapter, write notes and eamples for each lesson under its tab. Chapter 9 Factoring 473
3 Factors and Greatest Common Factors Vocabulary prime number composite number prime factorization factored form greatest common factor (GCF) Find prime factorizations of integers and monomials. Find the greatest common factors of integers and monomials. are prime numbers related to the search for etraterrestrial life? In the search for etraterrestrial life, scientists listen to radio signals coming from faraway galaies. How can they be sure that a particular radio signal was deliberately sent by intelligent beings instead of coming from some natural phenomenon? What if that signal began with a series of beeps in a pattern comprised of the first 30 prime numbers ( beepbeep, beepbeepbeep, and so on)? PRIME FACTORIZATION Recall that when two or more numbers are multiplied, each number is a factor of the product. Some numbers, like 8, can be epressed as the product of different pairs of whole numbers. This can be shown geometrically. Consider all of the possible rectangles with whole number dimensions that have areas of 8 square units The number 8 has 6 factors,, 2, 3, 6, 9, and 8. Whole numbers greater than can be classified by their number of factors. Words A whole number, greater than, whose only factors are and itself, is called a prime number. A whole number, greater than, that has more than two factors is called a composite number. Prime and Composite Numbers Eamples 2, 3, 5, 7,, 3, 7, 9 4, 6, 8, 9, 0, 2, 4, 5, 6, 8 0 and are neither prime nor composite. Study Tip Listing Factors Notice that in Eample, 6 is listed as a factor of 36 only once. Eample Classify Numbers as Prime or Composite Factor each number. Then classify each number as prime or composite. a. 36 To find the factors of 36, list all pairs of whole numbers whose product is Therefore, the factors of 36, in increasing order, are, 2, 3, 4, 6, 9, 2, 8, and 36. Since 36 has more than two factors, it is a composite number. 474 Chapter 9 Factoring
4 Study Tip Prime Numbers Before deciding that a number is prime, try dividing it by all of the prime numbers that are less than the square root of that number. b. 23 The only whole numbers that can be multiplied together to get 23 are and 23. Therefore, the factors of 23 are and 23. Since the only factors of 23 are and itself, 23 is a prime number. When a whole number is epressed as the product of factors that are all prime numbers, the epression is called the prime factorization of the number. Eample 2 Prime Factorization of a Positive Integer Find the prime factorization of 90. Method The least prime factor of 90 is The least prime factor of 45 is The least prime factor of 5 is 3. All of the factors in the last row are prime. Thus, the prime factorization of 90 is Method 2 Use a factor tree Study Tip Unique Factorization Theorem The prime factorization of every number is unique ecept for the order in which the factors are written and All of the factors in the last branch of the factor tree are prime. Thus, the prime factorization of 90 is or Usually the factors are ordered from the least prime factor to the greatest. A negative integer is factored completely when it is epressed as the product of and prime numbers. Eample 3 Prime Factorization of a Negative Integer Find the prime factorization of Epress 40 as times Thus, the prime factorization of 40 is or A monomial is in factored form when it is epressed as the product of prime numbers and variables and no variable has an eponent greater than. Lesson 9 Factors and Greatest Common Factors 475
5 Eample 4 Prime Factorization of a Monomial Factor each monomial completely. a. 2a 2 b 3 2a 2 b a a b b b a a b b b , a 2 a a, and b 3 b b b Thus, 2a 2 b 3 in factored form is a a b b b. b. 66pq 2 66pq 2 66 p q q Epress 66 as times p q q p q q 33 3 Thus, 66pq 2 in factored form is 2 3 p q q. GREATEST COMMON FACTOR Two or more numbers may have some common prime factors. Consider the prime factorization of 48 and Factor each number Circle the common prime factors. The integers 48 and 60 have two 2s and one 3 as common prime factors. The product of these common prime factors, or 2, is called the greatest common factor (GCF) of 48 and 60. The GCF is the greatest number that is a factor of both original numbers. Greatest Common Factor (GCF) The GCF of two or more integers is the product of the prime factors common to the integers. The GCF of two or more monomials is the product of their common factors when each monomial is in factored form. If two or more integers or monomials have a GCF of, then the integers or monomials are said to be relatively prime. Study Tip Alternative Method You can also find the greatest common factor by listing the factors of each number and finding which of the common factors is the greatest. Consider Eample 5a. 5:, 3, 5, 5 6:, 2, 4, 8, 6 The only common factor, and therefore, the greatest common factor, is. Eample 5 GCF of a Set of Monomials Find the GCF of each set of monomials. a. 5 and Factor each number Circle the common prime factors, if any. There are no common prime factors, so the GCF of 5 and 6 is. This means that 5 and 6 are relatively prime. b y and 54y 2 z 36 2 y y 54y 2 z y y z Factor each number. The GCF of 36 2 y and 54y 2 z is y or 8y. Circle the common prime factors. 476 Chapter 9 Factoring
6 Eample 6 Use Factors GEOMETRY The area of a rectangle is 28 square inches. If the length and width are both whole numbers, what is the maimum perimeter of the rectangle? Find the factors of 28, and draw rectangles with each length and width. Then find each perimeter. The factors of 28 are, 2, 4, 7, 4, and P or P or 32 P or 22 The greatest perimeter is 58 inches. The rectangle with this perimeter has a length of 28 inches and a width of inch. Concept Check Guided Practice GUIDED PRACTICE KEY. Determine whether the following statement is true or false. If false, provide a countereample. All prime numbers are odd. 2. Eplain what it means for two numbers to be relatively prime. 3. OPEN ENDED Name two monomials whose GCF is 5 2. Find the factors of each number. Then classify each number as prime or composite Find the prime factorization of each integer Factor each monomial completely. 0. 4p 2. 39b 3 c yz 2 Find the GCF of each set of monomials. 3. 0, y, 36y , 63, n, 2m 7. 2a 2 b, 90a 2 b 2 c 8. 5r 2, 35s 2, 70rs Application 9. GARDENING Ashley is planting 20 tomato plants in her garden. In what ways can she arrange them so that she has the same number of plants in each row, at least 5 rows of plants, and at least 5 plants in each row? Practice and Apply Find the factors of each number. Then classify each number as prime or composite Lesson 9 Factors and Greatest Common Factors 477
7 Homework Help For Eercises See Eamples 20 27, 62, 65, , , 5 63, , 67 6 Etra Practice See page 839. GEOMETRY For Eercises 28 and 29, consider a rectangle whose area is 96 square millimeters and whose length and width are both whole numbers. 28. What is the minimum perimeter of the rectangle? Eplain your reasoning. 29. What is the maimum perimeter of the rectangle? Eplain your reasoning. COOKIES For Eercises 30 and 3, use the following information. A bakery packages cookies in two sizes of boes, one with 8 cookies and the other with 24 cookies. A small number of cookies are to be wrapped in cellophane before they are placed in a bo. To save money, the bakery will use the same size cellophane packages for each bo. 30. How many cookies should the bakery place in each cellophane package to maimize the number of cookies in each package? 3. How many cellophane packages will go in each size bo? Find the prime factorization of each integer Factor each monomial completely d y a 3 b gh pq n 3 m yz a 2 bc 2 Find the GCF of each set of monomials , , , , , 20, , 63, a, 28b d 2, 30c 2 d gh, 36g 2 h p 2 q, 32r 2 t 58. 8, 30y, 54y a 2, 63a 3 b 2, 9b m 2 n 2, 8mn, 2m 2 n a 2 b, 96a 2 b 3, 28a 2 b NUMBER THEORY Twin primes are two consecutive odd numbers that are prime. The first pair of twin primes is 3 and 5. List the net five pairs of twin primes. Marching Bands Drum Corps International (DCI) is a nonprofit youth organization serving junior drum and bugle corps around the world. Members of these marching bands range from 4 to 2 years of age. Source: Chapter 9 Factoring MARCHING BANDS For Eercises 63 and 64, use the following information. Central High s marching band has 75 members, and the band from Northeast High has 90 members. During the halftime show, the bands plan to march into the stadium from opposite ends using formations with the same number of rows. 63. If the bands want to match up in the center of the field, what is the maimum number of rows? 64. How many band members will be in each row after the bands are combined? NUMBER THEORY For Eercises 65 and 66, use the following information. One way of generating prime numbers is to use the formula 2 p, where p is a prime number. Primes found using this method are called Mersenne primes. For eample, when p 2, The first Mersenne prime is Find the net two Mersenne primes. 66. Will this formula generate all possible prime numbers? Eplain your reasoning. Online Research Data Update What is the greatest known prime number? Visit to learn more.
8 67. GEOMETRY The area of a triangle is 20 square centimeters. What are possible wholenumber dimensions for the base and height of the triangle? Finding the GCF of distances will help you make a scale model of the solar system. Visit webquest to continue work on your WebQuest project. Standardized Test Practice 68. CRITICAL THINKING Suppose 6 is a factor of ab, where a and b are natural numbers. Make a valid argument to eplain why each assertion is true or provide a countereample to show that an assertion is false. a. 6 must be a factor of aorof b. b. 3 must be a factor of aorof b. c. 3 must be a factor of a and of b. 69. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. How are prime numbers related to the search for etraterrestrial life? Include the following in your answer: a list of the first 30 prime numbers and an eplanation of how you found them, and an eplanation of why a signal of this kind might indicate that an etraterrestrial message is to follow. 70. Miko claims that there are at least four ways to design a 20squarefoot rectangular space that can be tiled with foot by foot tiles. Which statement best describes this claim? A Her claim is false because 20 is a prime number. B Her claim is false because 20 is not a perfect square. C Her claim is true because 240 is a multiple of 20. D Her claim is true because 20 has at least eight factors. 7. Suppose Ψ is defined as the largest prime factor of. For which of the following values of would Ψ have the greatest value? A 53 B 74 C 99 D 7 Maintain Your Skills Mied Review Find each product. (Lessons 87 and 88) 72. (2 ) (3a 5)(3a 5) 74. (7p 2 4)(7p 2 4) 75. (6r 7)(2r 5) 76. (0h k)(2h 5k) 77. (b 4)(b 2 3b 8) Find the value of r so that the line that passes through the given points has the given slope. (Lesson 5) 78. (, 2), ( 2, r), m ( 5, 9), (r, 6), m RETAIL SALES A department store buys clothing at wholesale prices and then marks the clothing up 25% to sell at retail price to customers. If the retail price of a jacket is $79, what was the wholesale price? (Lesson 37) Getting Ready for the Net Lesson PREREQUISITE SKILL Use the Distributive Property to rewrite each epression. (To review the Distributive Property, see Lesson 5.) 8. 5(2 8) 82. a(3a ) 83. 2g(3g 4) 84. 4y(3y 6) 85. 7b 7c Lesson 9 Factors and Greatest Common Factors 479
9 A Preview of Lesson 92 Factoring Using the Distributive Property Sometimes you know the product of binomials and are asked to find the factors. This is called factoring. You can use algebra tiles and a product mat to factor binomials. Activity Use algebra tiles to factor 3 6. Model the polynomial 3 6. Arrange the tiles into a rectangle. The total area of the rectangle represents the product, and its length and width represent the factors. 2 3 The rectangle has a width of 3 and a length of 2. So, 3 6 3( 2). Activity 2 Use algebra tiles to factor 2 4. Model the polynomial 2 4. Arrange the tiles into a rectangle The rectangle has a width of and a length of 4. So, 2 4 ( 4). Model and Analyze Use algebra tiles to factor each binomial Tell whether each binomial can be factored. Justify your answer with a drawing MAKE A CONJECTURE Write a paragraph that eplains how you can use algebra tiles to determine whether a binomial can be factored. Include an eample of one binomial that can be factored and one that cannot. 480 Investigating SlopeIntercept Form 480 Chapter 9 Factoring
10 Factoring Using the Distributive Property Vocabulary factoring factoring by grouping Factor polynomials by using the Distributive Property. Solve quadratic equations of the form a 2 b 0. can you determine how long a baseball will remain in the air? Nolan Ryan, the greatest strikeout pitcher in the history of baseball, had a fastball clocked at 98 miles per hour or about 5 feet per second. If he threw a ball directly upward with the same velocity, the height h of the ball in feet above the point at which he released it could be modeled by the formula h 5t 6t 2, where t is the time in seconds. You can use factoring and the Zero Product Property to determine how long the ball would remain in the air before returning to his glove. Study Tip Look Back To review the Distributive Property, see Lesson 5. FACTOR BY USING THE DISTRIBUTIVE PROPERTY In Chapter 8, you used the Distributive Property to multiply a polynomial by a monomial. 2a(6a 8) 2a(6a) 2a(8) 2a 2 6a You can reverse this process to epress a polynomial as the product of a monomial factor and a polynomial factor. 2a 2 6a 2a(6a) 2a(8) 2a(6a 8) Thus, a factored form of 2a 2 6a is 2a(6a 8). Factoring a polynomial means to find its completely factored form. The epression 2a(6a 8) is not completely factored since 6a 8 can be factored as 2(3a 4). Eample Use the Distributive Property Use the Distributive Property to factor each polynomial. a. 2a 2 6a First, find the GCF of 2a 2 and 6a. 2a a a Factor each number. 6a a Circle the common prime factors. GCF: 2 2 a or 4a Write each term as the product of the GCF and its remaining factors. Then use the Distributive Property to factor out the GCF. 2a 2 6a 4a(3 a) 4a(2 2) Rewrite each term using the GCF. 4a(3a) 4a(4) Simplify remaining factors. 4a(3a 4) Distributive Property Thus, the completely factored form of 2a 2 6a is 4a(3a 4). Lesson 92 Factoring Using the Distributive Property 48
11 b. 8cd 2 2c 2 d 9cd 8cd c d d 2c 2 d c c d 9cd 3 3 c d GCF: 3 c d or 3cd Factor each number. Circle the common prime factors. 8cd 2 2c 2 d 9cd 3cd(6d) 3cd(4c) 3cd(3) 3cd(6d 4c 3) Rewrite each term using the GCF. Distributive Property Study Tip Factoring by Grouping Sometimes you can group terms in more than one way when factoring a polynomial. For eample, the polynomial in Eample 2 could have been factored in the following way. 4ab 8b 3a 6 (4ab 3a) (8b 6) a(4b 3) 2(4b 3) (4b 3)(a 2) Notice that this result is the same as in Eample 2. Study Tip Factoring Trinomials Since the order in which factors are multiplied does not affect the product, ( 5 3)(y 7) is also a correct factoring of 35 5y 3y 2. The Distributive Property can also be used to factor some polynomials having four or more terms. This method is called factoring by grouping because pairs of terms are grouped together and factored. The Distributive Property is then applied a second time to factor a common binomial factor. Eample 2 Use Grouping Factor 4ab 8b 3a 6. 4ab 8b 3a 6 (4ab 8b) (3a 6) Group terms with common factors. 4b(a 2) 3(a 2) Factor the GCF from each grouping. (a 2)(4b 3) Distributive Property CHECK Use the FOIL method. F O I L (a 2)(4b 3) (a)(4b) (a)(3) (2)(4b) (2)(3) 4ab 3a 8b 6 Recognizing binomials that are additive inverses is often helpful when factoring by grouping. For eample, 7 y and y 7 are additive inverses because their sum is 0. By rewriting 7 y in the factored form (y 7), factoring by grouping is made possible in the following eample. Eample 3 Use the Additive Inverse Property Factor 35 5y 3y y 3y 2 (35 5y) (3y 2) Group terms with common factors. 5(7 y) 3(y 7) Factor the GCF from each grouping. 5( )(y 7) 3(y 7) 7 y (y 7) 5(y 7) 3(y 7) 5( ) 5 (y 7)( 5 3) Distributive Property Factoring by Grouping Words A polynomial can be factored by grouping if all of the following situations eist. There are four or more terms. Terms with common factors can be grouped together. The two common factors are identical or are additive inverses of each other. Symbols a b ay by (a b) y(a b) (a b)( y) 482 Chapter 9 Factoring
12 SOLVE EQUATIONS BY FACTORING factoring. Consider the following products. Some equations can be solved by 6(0) 0 0( 3) 0 (5 5)(0) 0 2( 3 3) 0 Notice that in each case, at least one of the factors is zero. These eamples illustrate the. Zero Product Property Zero Product Property Words If the product of two factors is 0, then at least one of the factors must be 0. Symbols For any real numbers a and b, if ab 0, then either a 0, b 0, or both a and b equal zero. Eample 4 Solve an Equation in Factored Form Solve (d 5)(3d 4) 0. Then check the solutions. If (d 5)(3d 4) 0, then according to the Zero Product Property either d 5 0 or 3d 4 0. (d 5)(3d 4) 0 Original equation d 5 0 or 3d 4 0 Set each factor equal to zero. d 5 3d 4 Solve each equation. The solution set is 5, 4 3. d 4 3 CHECK Substitute 5 and 4 for d in the original equation. 3 (d 5)(3d 4) 0 (d 5)(3d 4) 0 (5 5)[3(5) 4] (0)(9) 0 9 (0) If an equation can be written in the form ab 0, then the Zero Product Property can be applied to solve that equation. Study Tip Common Misconception You may be tempted to try to solve the equation in Eample 5 by dividing each side of the equation by. Remember, however, that is an unknown quantity. If you divide by, you may actually be dividing by zero, which is undefined. Eample 5 Solve an Equation by Factoring Solve 2 7. Then check the solutions. Write the equation so that it is of the form ab Original equation Subtract 7 from each side. ( 7) 0 Factor the GCF of 2 and 7, which is. 0 or 7 0 Zero Product Property 7 Solve each equation. The solution set is {0, 7}. Check by substituting 0 and 7 for in the original equation. Lesson 92 Factoring Using the Distributive Property 483
13 Concept Check Guided Practice GUIDED PRACTICE KEY Application. Write as a product of factors in three different ways. Then decide which of the three is the completely factored form. Eplain your reasoning. 2. OPEN ENDED Give an eample of the type of equation that can be solved by using the Zero Product Property. 3. Eplain why ( 2)( 4) 0 cannot be solved by dividing each side by 2. Factor each polynomial z 40z m 2 np 2 36m 2 n 2 p 7. 2a 3 b 2 8ab 6a 2 b y 2 5y 4y c 0c 2 2d 4cd Solve each equation. Check your solutions. 0. h(h 5) 0. (n 4)(n 2) m 3m 2 PHYSICAL SCIENCE For Eercises 3 5, use the information below and in the graphic. A flare is launched from a life raft. The height h of the flare in feet above the sea is modeled by the formula h 00t 6t 2, where t is the time in seconds after the flare is launched. 3. At what height is the flare when it returns to the sea? 4. Let h 0 in the equation h 00t 6t 2 and solve for t. 5. How many seconds will it take for the flare to return to the sea? Eplain your reasoning. 00 ft/s h 00t 6t 2 h 0 Practice and Apply Homework Help For Eercises See Eamples 6 29, , , 5 Etra Practice See page 840. Factor each polynomial y 7. 6a 4b 8. a 5 b a 9. 3 y cd 3d 2. 4gh 8h 22. 5a 2 y 30ay 23. 8bc 2 24bc y 2 z 40y 3 z a 2 bc 2 48abc a a 2 b 2 a 3 b y 2 25y 28. 2a 3 20b 2 32c 29. 3p 3 q 9pq 2 36pq y 2 9y 8y a 2 5a 8a a 3ay 4b 3by 37. 2my 7 7m 2y 38. 8a 6 2a y 5 2y 484 Chapter 9 Factoring GEOMETRY For Eercises 40 and 4, use the following information. A quadrilateral has 4 sides and 2 diagonals. A pentagon has 5 sides and 5 diagonals. You can use 2 n2 3 n to find the number of diagonals in a polygon with n sides Write this epression in factored form. 4. Find the number of diagonals in a decagon (0sided polygon).
14 SOFTBALL For Eercises 42 and 43, use the following information. Albertina is scheduling the games for a softball league. To find the number of games she needs to schedule, she uses the equation g 2 n2 n, where g 2 represents the number of games needed for each team to play each other team eactly once and n represents the number of teams. 42. Write this equation in factored form. 43. How many games are needed for 7 teams to play each other eactly 3 times? GEOMETRY Write an epression in factored form for the area of each shaded region b 2 2 a r r 2 GEOMETRY Find an epression for the area of a square with the given perimeter. 46. P 2 20y in. 47. P 36a 6b cm Solve each equation. Check your solutions. 48. ( 24) a(a 6) (q 4)(3q 5) 0 5. (3y 9)(y 7) (2b 3)(3b 8) (4n 5)(3n 7) z 2 2z d 2 35d MARINE BIOLOGY In a pool at a water park, a dolphin jumps out of the water traveling at 20 feet per second. Its height h, in feet, above the water after t seconds is given by the formula h 20t 6t 2. How long is the dolphin in the air before returning to the water? Marine Biologist Marine biologists study factors that affect organisms living in and near the ocean. Online Research For information about a career as a marine biologist, visit: careers Source: National Sea Grant Library 6. BASEBALL Malik popped a ball straight up with an initial upward velocity of 45 feet per second. The height h, in feet, of the ball above the ground is modeled by the equation h 2 48t 6t 2. How long was the ball in the air if the catcher catches the ball when it is 2 feet above the ground? 62. CRITICAL THINKING Factor a y a b y a y b b y. 63. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. How can you determine how long a baseball will remain in the air? Include the following in your answer: an eplanation of how to use factoring and the Zero Product Property to find how long the ball would be in the air, and an interpretation of each solution in the contet of the problem. Lesson 92 Factoring Using the Distributive Property 485
15 Standardized Test Practice 64. The total number of feet in yards, y feet, and z inches is z A 3 y. B 2( y z). 2 y C 3y 36z. D z QUANTITATIVE COMPARISON Compare the quantity in Column A and the quantity in Column B. Then determine whether: A B C D the quantity in Column A is greater, the quantity in Column B is greater, the two quantities are equal, or the relationship cannot be determined from the information given. Column A Column B the negative solution of the negative solution of (a 2)(a 5) 0 (b 6)(b ) 0 Maintain Your Skills Mied Review Factor each number. Then classify each number as prime or composite. (Lesson 9) Find each product. (Lesson 88) 69. (4s 3 3) (2p 5q)(2p 5q) 7. (3k 8)(3k 8) Simplify. Assume that no denominator is equal to zero. (Lesson 82) s s y 34p7q2r 5 22y ( p3 qr ) FINANCE Michael uses at most 60% of his annual FlynnCo stock dividend to purchase more shares of FlynnCo stock. If his dividend last year was $885 and FlynnCo stock is selling for $4 per share, what is the greatest number of shares that he can purchase? (Lesson 62) Getting Ready for the Net Lesson PREREQUISITE SKILL Find each product. (To review multiplying polynomials, see Lesson 87.) 76. (n 8)(n 3) 77. ( 4)( 5) 78. (b 0)(b 7) 79. (3a )(6a 4) 80. (5p 2)(9p 3) 8. (2y 5)(4y 3) P ractice Quiz Lessons 9 and 92. Find the factors of 225. Then classify the number as prime or composite. (Lesson 9) 2. Find the prime factorization of 320. (Lesson 9) 3. Factor 78a 2 bc 3 completely. (Lesson 9) 4. Find the GCF of 54 3, 42 2 y, and 30y 2. (Lesson 9) Factor each polynomial. (Lesson 92) 5. 4y 2 y 6. 32a 2 b 40b 3 8a 2 b py 6p 5y 40 Solve each equation. Check your solutions. (Lesson 92) 8. (8n 5)(n 4) Chapter 9 Factoring
16 A Preview of Lesson 93 Factoring Trinomials You can use algebra tiles to factor trinomials. If a polynomial represents the area of a rectangle formed by algebra tiles, then the rectangle s length and width are factors of the area. Activity Use algebra tiles to factor Model the polynomial Place the 2 tile at the corner of the product mat. Arrange the tiles into a rectangular array. Because 5 is prime, the 5 tiles can be arranged in a rectangle in one way, a by5 rectangle. 2 Complete the rectangle with the tiles. 5 The rectangle has a width of and a length of 5. Therefore, ( )( 5). 2 Activity 2 Use algebra tiles to factor Model the polynomial Place the 2 tile at the corner of the product mat. Arrange the tiles into a rectangular array. Since 6 2 3, try a 2by3 rectangle. Try to complete the rectangle. Notice that there are two etra tiles. 2 (continued on the net page) Algebra Activity Factoring Trinomials 487
17 Algebra Activity Arrange the tiles into a by6 rectangular array. This time you can complete the rectangle with the tiles. The rectangle has a width of and a length of 6. Therefore, ( )( 6). 6 2 Activity 3 Use algebra tiles to factor Model the polynomial Place the 2 tile at the corner of the product mat. Arrange the tiles into a by3 rectangular array as shown. 2 Place the tile as shown. Recall that you can add zeropairs without changing the value of the polynomial. In this case, add a zero pair of tiles zero pair The rectangle has a width of and a length of 3. Therefore, ( )( 3). Model Use algebra tiles to factor each trinomial Investigating SlopeIntercept Form 488 Chapter 9 Factoring
18 Factoring Trinomials: 2 b c Factor trinomials of the form 2 b c. Solve equations of the form 2 b c 0. can factoring be used to find the dimensions of a garden? Tamika has enough bricks to make a 30foot border around the rectangular vegetable garden she is planting. The booklet she got from the nursery says that the plants will need a space of 54 square feet to grow. What should the dimensions of her garden be? To solve this problem, you need to find two numbers whose product is 54 and whose sum is 5, half the perimeter of the garden. A 54 ft 2 P 30 ft FACTOR 2 b c In Lesson 9, you learned that when two numbers are multiplied, each number is a factor of the product. Similarly, when two binomials are multiplied, each binomial is a factor of the product. To factor some trinomials, you will use the pattern for multiplying two binomials. Study the following eample. F O I L ( 2)( 3) ( ) ( 3) ( 2) (2 3) Use the FOIL method Simplify. 2 (3 2) 6 Distributive Property Simplify. Observe the following pattern in this multiplication. Study Tip Reading Math A quadratic trinomial is a trinomial of degree 2. This means that the greatest eponent of the variable is 2. Words ( 2)( 3) 2 (3 2) (2 3) ( m)( n) 2 (n m) mn 2 (m n) mn 2 b c b m n and c mn Notice that the coefficient of the middle term is the sum of m and n and the last term is the product of m and n. This pattern can be used to factor quadratic trinomials of the form 2 b c. Factoring 2 b c To factor quadratic trinomials of the form 2 b c, find two integers, m and n, whose sum is equal to b and whose product is equal to c. Then write 2 b c using the pattern ( m)( n). Symbols 2 b c ( m)( n) when m n b and mn c. Eample ( 2)( 3), since and Lesson 93 Factoring Trinomials: 2 b c 489
19 To determine m and n, find the factors of c and use a guessandcheck strategy to find which pair of factors has a sum of b. Eample Factor b and c Are Positive In this trinomial, b 6 and c 8. You need to find two numbers whose sum is 6 and whose product is 8. Make an organized list of the factors of 8, and look for the pair of factors whose sum is 6. Factors of 8 Sum of Factors, 8 9 2, 4 6 The correct factors are 2 and ( m)( n) Write the pattern. ( 2)( 4) m 2 and n 4 CHECK You can check this result by multiplying the two factors. F O I L ( 2)( 4) FOIL method Simplify. When factoring a trinomial where b is negative and c is positive, you can use what you know about the product of binomials to help narrow the list of possible factors. Study Tip Testing Factors Once you find the correct factors, there is no need to test any other factors. Therefore, it is not necessary to test 4 and 4 in Eample 2. Eample 2 Factor b Is Negative and c Is Positive In this trinomial, b 0 and c 6. This means that m n is negative and mn is positive. So m and n must both be negative. Therefore, make a list of the negative factors of 6, and look for the pair of factors whose sum is 0. Factors of 6 Sum of Factors, 6 7 2, 8 0 4, ( m)( n) ( 2)( 8) The correct factors are 2 and 8. Write the pattern. m 2 and n 8 TEACHING TIP CHECK You can check this result by using a graphing calculator. Graph y and y ( 2)( 8) on the same screen. Since only one graph appears, the two graphs must coincide. Therefore, the trinomial has been factored correctly. [ 0, 0] scl: by [ 0, 0] scl: You will find that keeping an organized list of the factors you have tested is particularly important when factoring a trinomial like 2 2, where the value of c is negative. 490 Chapter 9 Factoring
20 Study Tip Alternate Method You can use the opposite of FOIL to factor trinomials. For instance, consider Eample ( )( ) Try factor pairs of 2 until the sum of the products of the Inner and Outer terms is. Eample 3 Factor 2 2. b Is Positive and c Is Negative In this trinomial, b and c 2. This means that m n is positive and mn is negative. So either m or n is negative, but not both. Therefore, make a list of the factors of 2, where one factor of each pair is negative. Look for the pair of factors whose sum is. Factors of 2 Sum of Factors, 2, 2 2, 6 4 2, 6 4 3, 4 3, 4 The correct factors are 3 and ( m)( n) Write the pattern. ( 3)( 4) m 3 and n 4 Eample 4 Factor b Is Negative and c Is Negative Since b 7 and c 8, m n is negative and mn is negative. So either m or n is negative, but not both. Factors of 8 Sum of Factors, 8 7, 8 7 2, 9 7 The correct factors are 2 and ( m)( n) ( 2)( 9) Write the pattern. m 2 and n 9 SOLVE EQUATIONS BY FACTORING Some equations of the form 2 b c 0 can be solved by factoring and then using the Zero Product Property. Eample 5 Solve an Equation by Factoring Solve Check your solutions Original equation Rewrite the equation so that one side equals 0. ( )( 6) 0 Factor. 0 or 6 0 Zero Product Property 6 Solve each equation. The solution set is {, 6}. CHECK Substitute and 6 for in the original equation () 2 5() 6 ( 6) 2 5( 6) Lesson 93 Factoring Trinomials: 2 b c 49
21 Eample 6 Solve a RealWorld Problem by Factoring YEARBOOK DESIGN A sponsor for the school yearbook has asked that the length and width of a photo in their ad be increased by the same amount in order to double the area of the photo. If the photo was originally 2 centimeters wide by 8 centimeters long, what should the new dimensions of the enlarged photo be? 2 8 Eplore Plan Begin by making a diagram like the one shown above, labeling the appropriate dimensions. Let the amount added to each dimension of the photo. The new length times the new width equals the new area (8)(2) Solve ( 2)( 8) 2(8)(2) Write the equation Multiply Subtract 92 from each side. ( 24)( 4) 0 Factor. old area Eamine 24 0 or 4 0 Zero Product Property 24 4 Solve each equation. The solution set is { 24, 4}. Only 4 is a valid solution, since dimensions cannot be negative. Thus, the new length of the photo should be 4 2 or 6 centimeters, and the new width should be 4 8 or 2 centimeters. Concept Check. Eplain why, when factoring 2 6 9, it is not necessary to check the sum of the factor pairs and 9 or 3 and OPEN ENDED Give an eample of an equation that can be solved using the factoring techniques presented in this lesson. Then, solve your equation. 3. FIND THE ERROR Peter and Aleta are solving GUIDED PRACTICE KEY Peter = 5 ( + 2) = 5 = 5 or + 2 = 5 = 3 Who is correct? Eplain your reasoning. Aleta = = 0 (  3)( + 5) = 03 = 0 or + 5 = 0 = 3 = 5 Guided Practice Factor each trinomial c 2 3c 2 6. n 2 3n p 2 2p a a y 3y Chapter 9 Factoring
22 Solve each equation. Check your solutions. 0. n 2 7n 6 0. a 2 5a p 2 9p y 2 9 0y d 2 3d 70 Application 6. NUMBER THEORY Find two consecutive integers whose product is 56. Practice and Apply Homework Help For Eercises See Eamples , 6 6, 62 Etra Practice See page 840. Factor each trinomial. 7. a 2 8a c 2 2c y 2 3y m 2 22m d 2 7d p 2 7p g 2 9g b 2 b h 2 3h n 2 3n y 2 y z 2 8z w w a 2 5ab 4b y 36y 2 GEOMETRY Find an epression for the perimeter of a rectangle with the given area. 35. area area Solve each equation. Check your solutions b 2 20b y 2 4y d 2 2d a 2 3a g 2 4g m 2 9m n 2 22n z 2 8 7z 46. h 2 5 6h k 2 0k c c 50. y 2 29y p p SUPREME COURT When the Justices of the Supreme Court assemble to go on the Bench each day, each Justice shakes hands with each of the other Justices for a total of 36 handshakes. The total number of handshakes h possible for n people is given by h n2 n. Write and solve an equation to determine the 2 number of Justices on the Supreme Court. 55. NUMBER THEORY Find two consecutive even integers whose product is GEOMETRY The triangle has an area of 40 square centimeters. Find the height h of the triangle. h cm Supreme Court The Conference handshake has been a tradition since the late 9th century. Source: (2h 6) cm CRITICAL THINKING Find all values of k so that each trinomial can be factored using integers k k k, k k, k 0 RUGBY For Eercises 6 and 62, use the following information. The length of a Rugby League field is 52 meters longer than its width w. 6. Write an epression for the area of the field. 62. The area of a Rugby League field is 860 square meters. Find the dimensions of the field. Lesson 93 Factoring Trinomials: 2 b c 493
23 63. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. How can factoring be used to find the dimensions of a garden? Include the following in your answer: a description of how you would find the dimensions of the garden, and an eplanation of how the process you used is related to the process used to factor trinomials of the form 2 b c. Standardized Test Practice 64. Which is the factored form of ? A ( )(y 42) B ( 2)( 2) C ( 3)( 4) D ( 6)( 7) 65. GRID IN What is the positive solution of p 2 3p 30 0? Graphing Calculator Use a graphing calculator to determine whether each factorization is correct. Write yes or no. If no, state the correct factorization ( 6)( 8) ( 5)( 2) ( 33)( 2) ( 0)( 2) Maintain Your Skills Mied Review Solve each equation. Check your solutions. (Lesson 92) 70. ( 3)(2 5) 0 7. b(7b 4) y 2 9y Find the GCF of each set of monomials. (Lesson 9) , 36, p 2 q 5, 2p 3 q y 5, 20 2 y 7, 75 3 y 4 INTERNET For Eercises 76 and 77, use the graph at the right. (Lessons 37 and 83) 76. Find the percent increase in the number of domain registrations from 997 to Use your answer from Eercise 76 to verify the claim that registrations grew more than 8fold from 997 to 2000 is correct. USA TODAY Snapshots Number of domain registrations climbs During the past four years,.com,.net and.org domain registrations have grown more than 8fold: 28.2 million 9 million million million Source: Network Solutions (VeriSign) By Cindy Hall and Bob Laird, USA TODAY Getting Ready for the Net Lesson PREREQUISITE SKILL Factor each polynomial. (To review factoring by grouping, see Lesson 92.) 78. 3y 2 2y 9y a 2 2a 2a p 2 6p 7p b 2 7b 2b g 2 2g 6g Chapter 9 Factoring
24 Factoring Trinomials: a 2 b c Vocabulary prime polynomial Factor trinomials of the form a 2 b c. Solve equations of the form a 2 b c 0. can algebra tiles be used to factor ? The factors of are the dimensions of the rectangle formed by the algebra tiles shown below. 2 2 The process you use to form the rectangle is the same mental process you can use to factor this trinomial algebraically. FACTOR a 2 b c For trinomials of the form 2 b c, the coefficient of 2 is. To factor trinomials of this form, you find the factors of c whose sum is b. We can modify this approach to factor trinomials whose leading coefficient is not. Study Tip Look Back To review factoring by grouping, see Lesson 92. F O I L (2 5)(3 ) Use the FOIL method Observe the following pattern in this product a 2 m n c a 2 b c and m n b and mn ac You can use this pattern and the method of factoring by grouping to factor Find two numbers, m and n, whose product is 6 5 or 30 and whose sum is 7. Factors of 30 Sum of Factors, , 5 7 The correct factors are 2 and m n 5 Write the pattern m 2 and n 5 (6 2 2) (5 5) Group terms with common factors. 2(3 ) 5(3 ) Factor the GCF from each grouping. (3 )(2 5) 3 is the common factor. Therefore, (3 )(2 5). Lesson 94 Factoring Trinomials: a 2 b c 495
25 Eample Factor a 2 b c a. Factor In this trinomial, a 7, b 22 and c 3. You need to find two numbers whose sum is 22 and whose product is 7 3 or 2. Make an organized list of the factors of 2 and look for the pair of factors whose sum is 22. Factors of 2 Sum of Factors, 2 22 The correct factors are and m n 3 Write the pattern m and n 2 (7 2 ) (2 3) Group terms with common factors. (7 ) 3(7 ) Factor the GCF from each grouping. (7 )( 3) Distributive Property CHECK You can check this result by multiplying the two factors. F O I L (7 )( 3) FOIL method Simplify. b. Factor In this trinomial, a 0, b 43 and c 28. Since b is negative, m n is negative. Since c is positive, mn is positive. So m and n must both be negative. Therefore, make a list of the negative factors of 0 28 or 280, and look for the pair of factors whose sum is 43. Study Tip Finding Factors Factor pairs in an organized list so you do not miss any possible pairs of factors. Factors of 280 Sum of Factors, , , , , , The correct factors are 8 and m n ( 8) ( 35) 28 (0 2 8) ( 35 28) 2(5 4) 7( 5 4) Write the pattern. m 8 and n 35 Group terms with common factors. Factor the GCF from each grouping. 2(5 4) 7( )(5 4) 5 4 ( )(5 4) 2(5 4) ( 7)(5 4) 7( ) 7 (5 4)(2 7) Distributive Property Sometimes the terms of a trinomial will contain a common factor. In these cases, first use the Distributive Property to factor out the common factor. Then factor the trinomial. Eample 2 Factor When a, b, and c Have a Common Factor Factor Notice that the GCF of the terms 3 2, 24, and 45 is 3. When the GCF of the terms of a trinomial is an integer other than, you should first factor out this GCF ( 2 8 5) Distributive Property 496 Chapter 9 Factoring
26 Study Tip Factoring Completely Always check for a GCF first before trying to factor a trinomial. Now factor Since the lead coefficient is, find two factors of 5 whose sum is 8. Factors of 5 Sum of Factors, 5 6 3, 5 8 The correct factors are 2 and 5. So, ( 3)( 5). Thus, the complete factorization of is 3( 3)( 5). A polynomial that cannot be written as a product of two polynomials with integral coefficients is called a prime polynomial. Eample 3 Determine Whether a Polynomial Is Prime Factor In this trinomial, a 2, b 5 and c 2. Since b is positive, m n is positive. Since c is negative, mn is negative. So either m or n is negative, but not both. Therefore, make a list of the factors of 2 2 or 4, where one factor in each pair is negative. Look for a pair of factors whose sum is 5. Factors of 4 Sum of Factors, 4 3, 4 3 2, 2 0 There are no factors whose sum is 5. Therefore, cannot be factored using integers. Thus, is a prime polynomial. SOLVE EQUATIONS BY FACTORING Some equations of the form a 2 b c 0 can be solved by factoring and then using the Zero Product Property. Eample 4 Solve Equations by Factoring Solve 8a 2 9a 5 4 3a. Check your solutions. 8a 2 9a 5 4 3a Original equation 8a 2 6a 9 0 Rewrite so that one side equals 0. (4a 3)(2a 3) 0 Factor the left side. 4a 3 0 or 2a 3 0 Zero Product Property 4a 3 2a 3 Solve each equation. a 3 4 The solution set is 3 4, 3 2. a 3 2 CHECK Check each solution in the original equation. 8a 2 9a 5 4 3a a 2 9a 5 4 3a Lesson 94 Factoring Trinomials: a 2 b c 497
27 Study Tip Factoring When a Is Negative When factoring a trinomial of the form a 2 b c, where a is negative, it is helpful to factor out a negative monomial. A model for the vertical motion of a projected object is given by the equation h 6t 2 vt s, where h is the height in feet, t is the time in seconds, v is the initial upward velocity in feet per second, and s is the starting height of the object in feet. Eample 5 Solve RealWorld Problems by Factoring PEP RALLY At a pep rally, small foam footballs are launched by cheerleaders using a slingshot. How long is a football in the air if a student in the stands catches it on its way down 26 feet above the gym floor? Use the model for vertical motion. h 6t 2 vt s Vertical motion model 26 6t 2 42t 6 h 26, v 42, s 6 0 6t 2 42t 20 Subtract 26 from each side. 0 2(8t 2 2t 0) Factor out t 2 2t 0 Divide each side by 2. 0 (8t 5)(t 2) Factor 8t 2 2t 0. 8t 5 0 or t 2 0 Zero Product Property 8t 5 t 2 Solve each equation. t 5 8 Height of release 6 ft t 0 v 42 ft/s Height of reception 26 ft The solutions are 5 8 second and 2 seconds. The first time represents how long it takes the football to reach a height of 26 feet on its way up. The later time represents how long it takes the ball to reach a height of 26 feet again on its way down. Thus, the football will be in the air for 2 seconds before the student catches it. Concept Check. Eplain how to determine which values should be chosen for m and n when factoring a polynomial of the form a 2 b c. 2. OPEN ENDED Write a trinomial that can be factored using a pair of numbers whose sum is 9 and whose product is FIND THE ERROR Dasan and Craig are factoring GUIDED PRACTICE KEY Dasan Factors of 8 Sum, 8 9 3, 6 9 9, = 2( ) = 2( + 9)( + 2) Craig Factors of 36 Sum, , , 2 5 4, 9 3 6, is prime. Guided Practice Who is correct? Eplain your reasoning. Factor each trinomial, if possible. If the trinomial cannot be factored using integers, write prime. 4. 3a 2 8a a 2 a p 2 4p n 2 4n Chapter 9 Factoring
28 Solve each equation. Check your solutions p 2 9p n 2 7n 20 Application 3. GYMNASTICS When a gymnast making a vault leaves the horse, her feet are 8 feet above the ground traveling with an initial upward velocity of 8 feet per second. Use the model for vertical motion to find the time t in seconds it takes for the gymnast s feet to reach the mat. (Hint: Let h 0, the height of the mat.) 8 ft 8 ft/s Practice and Apply Homework Help For Eercises See Eamples Etra Practice See page 840. Factor each trinomial, if possible. If the trinomial cannot be factored using integers, write prime p 2 5p d 2 6d k 2 9k g 2 2g a 2 9a c 2 7c p 2 25p y 2 6y n 2 n z 2 7z r 2 4r y 25y a 2 9ab 0b 2 CRITICAL THINKING Find all values of k so that each trinomial can be factored as two binomials using integers k k k, k 0 Cliff Diving In Acapulco, Meico, divers leap from La Quebrada, the Break in the Rocks, diving headfirst into the Pacific Ocean 05 feet below. Source: acapulcotravel. web.com.m Solve each equation. Check your solutions n 2 25n a 2 3a t 2 3 t (3y 2)(y 3) y (4a )(a 2) 7a 5 GEOMETRY For Eercises 49 and 50, use the following information. A rectangle with an area of 35 square inches is formed by cutting off strips of equal width from a rectangular piece of paper. 49. Find the width of each strip. 50. Find the dimensions of the new rectangle. 7 in. 9 in CLIFF DIVING Suppose a diver leaps from the edge of a cliff 80 feet above the ocean with an initial upward velocity of 8 feet per second. How long it will take the diver to enter the water below? Lesson 94 Factoring Trinomials: a 2 b c 499
29 52. CLIMBING Damaris launches a grappling hook from a height of 6 feet with an initial upward velocity of 56 feet per second. The hook just misses the stone ledge of a building she wants to scale. As it falls, the hook anchors on the ledge, which is 30 feet above the ground. How long was the hook in the air? Standardized Test Practice 53. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. How can algebra tiles be used to factor ? Include the following in your answer: the dimensions of the rectangle formed, and an eplanation, using words and drawings, of how this geometric guessandcheck process of factoring is similar to the algebraic process described on page What are the solutions of 2p 2 p 3 0? A 2 3 and B 2 3 and C 3 2 and D 3 and 2 Maintain Your Skills Mied Review 55. Suppose a person standing atop a building 398 feet tall throws a ball upward. If the person releases the ball 4 feet above the top of the building, the ball s height h, in feet, after t seconds is given by the equation h 6t 2 48t 402. After how many seconds will the ball be 338 feet from the ground? A 3.5 B 4 C 4.5 D 5 Factor each trinomial, if possible. If the trinomial cannot be factored using integers, write prime. (Lesson 93) 56. a 2 4a t 2 2t d 2 5d 44 Solve each equation. Check your solutions. (Lesson 92) 59. (y 4)(5y 7) (2k 9)(3k 2) u u BUSINESS Jake s Garage charges $83 for a twohour repair job and $85 for a fivehour repair job. Write a linear equation that Jake can use to bill customers for repair jobs of any length of time. (Lesson 53) Getting Ready for the Net Lesson PREREQUISITE SKILL Find the principal square root of each number. (To review square roots, see Lesson 27.) P ractice Quiz 2 Factor each trinomial, if possible. If the trinomial cannot be factored using integers, write prime. (Lessons 93 and 94) p 2 6p a 2 24a 5 4. n 2 7n c 2 62c y 2 33y 54 Solve each equation. Check your solutions. (Lessons 93 and 94) 7. b 2 4b y 2 7y a 2 25a 4 Lessons 93 and Chapter 9 Factoring
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 92 through 96 Factor polynomials.
More informationFactoring. 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
More information6706_PM10SB_C4_CO_pp192193.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
More informationA.3. Polynomials and Factoring. Polynomials. What you should learn. Definition of a Polynomial in x. Why you should learn it
Appendi A.3 Polynomials and Factoring A23 A.3 Polynomials and Factoring What you should learn Write polynomials in standard form. Add,subtract,and multiply polynomials. Use special products to multiply
More informationThe majority of college students hold credit cards. According to the Nellie May
CHAPTER 6 Factoring Polynomials 6.1 The Greatest Common Factor and Factoring by Grouping 6. Factoring Trinomials of the Form b c 6.3 Factoring Trinomials of the Form a b c and Perfect Square Trinomials
More informationSPECIAL PRODUCTS AND FACTORS
CHAPTER 442 11 CHAPTER TABLE OF CONTENTS 111 Factors and Factoring 112 Common Monomial Factors 113 The Square of a Monomial 114 Multiplying the Sum and the Difference of Two Terms 115 Factoring the
More informationReview 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
More informationSECTION P.5 Factoring Polynomials
BLITMCPB.QXP.0599_4874 /0/0 0:4 AM Page 48 48 Chapter P Prerequisites: Fundamental Concepts of Algebra Technology Eercises Critical Thinking Eercises 98. The common cold is caused by a rhinovirus. The
More informationFactoring 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
More informationPolynomial 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
More informationVeterans 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
More informationSUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills
SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)
More informationPolynomials and Factoring
7.6 Polynomials and Factoring Basic Terminology A term, or monomial, is defined to be a number, a variable, or a product of numbers and variables. A polynomial is a term or a finite sum or difference of
More informationFactoring Polynomials
Factoring Polynomials 8A Factoring Methods 81 Factors and Greatest Common Factors Lab Model Factoring 82 Factoring by GCF Lab Model Factorization of Trinomials 83 Factoring x 2 + bx + c 84 Factoring
More informationSummer Math Exercises. For students who are entering. PreCalculus
Summer Math Eercises For students who are entering PreCalculus 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
More informationFACTORING ax 2 bx c WITH a 1
296 (6 20) Chapter 6 Factoring 6.4 FACTORING a 2 b c WITH a 1 In this section The ac Method Trial and Error Factoring Completely In Section 6.3 we factored trinomials with a leading coefficient of 1. In
More informationPOLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
More information9.3 OPERATIONS WITH RADICALS
9. Operations with Radicals (9 1) 87 9. OPERATIONS WITH RADICALS In this section Adding and Subtracting Radicals Multiplying Radicals Conjugates In this section we will use the ideas of Section 9.1 in
More informationDefinitions 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
More informationA Quick Algebra Review
1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals
More information7.2 Quadratic Equations
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
More information10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED
CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations
More informationFACTORING 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
More informationHow To Factor By Gcf In Algebra 1.5
72 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
More informationPolynomials. Polynomials
Preview of Algebra 1 Polynomials 1A Introduction to Polynomials 11 Polynomials LAB Model Polynomials 1 Simplifying Polynomials 1B Polynomial Operations LAB Model Polynomial Addition 13 Adding Polynomials
More informationFactoring 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
More informationPolynomials. 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
More informationFactor 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
More informationPolynomial 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
More informationFactoring Polynomials
Factoring Polynomials 8A Factoring Methods 81 Factors and Greatest Common Factors Lab Model Factorization by GCF 82 Factoring by GCF Lab Model Factorization of x 2 + bx + c 83 Factoring x 2 + bx + c
More informationMath 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.
Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used
More informationSection 5.0A Factoring Part 1
Section 5.0A Factoring Part 1 I. Work Together A. Multiply the following binomials into trinomials. (Write the final result in descending order, i.e., a + b + c ). ( 7)( + 5) ( + 7)( + ) ( + 7)( + 5) (
More informationLesson 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
More informationCPM Educational Program
CPM Educational Program A California, NonProfit Corporation Chris Mikles, National Director (888) 8084276 email: mikles @cpm.org CPM Courses and Their Core Threads Each course is built around a few
More informationUsing the Area Model to Teach Multiplying, Factoring and Division of Polynomials
visit us at www.cpm.org Using the Area Model to Teach Multiplying, Factoring and Division of Polynomials For more information about the materials presented, contact Chris Mikles mikles@cpm.org From CCA
More informationFactors 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
More information15.1 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
More informationHow To Solve Factoring Problems
05W4801AM1.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
More information1.3 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.
More informationMATH 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
More information1.1 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)
More information5.1 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.
More informationAnswers to Basic Algebra Review
Answers to Basic Algebra Review 1. 1.1 Follow the sign rules when adding and subtracting: If the numbers have the same sign, add them together and keep the sign. If the numbers have different signs, subtract
More informationLesson 9.1 Solving Quadratic Equations
Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One intercept and all nonnegative yvalues. b. The verte in the third quadrant and no intercepts. c. The verte
More informationAlum Rock Elementary Union School District Algebra I Study Guide for Benchmark III
Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Name Date Adding and Subtracting Polynomials Algebra Standard 10.0 A polynomial is a sum of one ore more monomials. Polynomial
More informationCopy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.
Algebra 2  Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers  {1,2,3,4,...}
More informationFive 5. Rational Expressions and Equations C H A P T E R
Five C H A P T E R Rational Epressions and Equations. Rational Epressions and Functions. Multiplication and Division of Rational Epressions. Addition and Subtraction of Rational Epressions.4 Comple Fractions.
More informationexpression 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
More informationMATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) offered through BSE (Basic Skills Education)
MATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,
More informationBig Bend Community College. Beginning Algebra MPC 095. Lab Notebook
Big Bend Community College Beginning Algebra MPC 095 Lab Notebook Beginning Algebra Lab Notebook by Tyler Wallace is licensed under a Creative Commons Attribution 3.0 Unported License. Permissions beyond
More informationFACTORING QUADRATICS 8.1.1 through 8.1.4
Chapter 8 FACTORING QUADRATICS 8.. through 8..4 Chapter 8 introduces students to rewriting quadratic epressions and solving quadratic equations. Quadratic functions are any function which can be rewritten
More informationcalled and explain why it cannot be factored with algebra tiles? and explain why it cannot be factored with algebra tiles?
Factoring Reporting Category Topic Expressions and Operations Factoring polynomials Primary SOL A.2c The student will perform operations on polynomials, including factoring completely first and seconddegree
More informationPolynomial and Synthetic Division. Long Division of Polynomials. Example 1. 6x 2 7x 2 x 2) 19x 2 16x 4 6x3 12x 2 7x 2 16x 7x 2 14x. 2x 4.
_.qd /7/5 9: AM Page 5 Section.. Polynomial and Synthetic Division 5 Polynomial and Synthetic Division What you should learn Use long division to divide polynomials by other polynomials. Use synthetic
More informationMathematics More Visual Using Algebra Tiles
www.cpm.org Chris Mikles CPM Educational Program A California Nonprofit Corporation 33 Noonan Drive Sacramento, CA 958 (888) 80876 fa: (08) 7778605 email: mikles@cpm.org An Eemplary Mathematics Program
More information1.3 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,
More informationFACTORING POLYNOMIALS
296 (540) 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
More informationFlorida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies  Upper
Florida Math 0028 Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies  Upper Exponents & Polynomials MDECU1: Applies the order of operations to evaluate algebraic
More informationESSENTIAL 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?
More informationFactoring 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
More informationUnit 6: Polynomials. 1 Polynomial Functions and End Behavior. 2 Polynomials and Linear Factors. 3 Dividing Polynomials
Date Period Unit 6: Polynomials DAY TOPIC 1 Polynomial Functions and End Behavior Polynomials and Linear Factors 3 Dividing Polynomials 4 Synthetic Division and the Remainder Theorem 5 Solving Polynomial
More informationSect 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
More information1) (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
More informationMATH 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
More informationAlgebra 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,
More informationPreCalculus II Factoring and Operations on Polynomials
Factoring... 1 Polynomials...1 Addition of Polynomials... 1 Subtraction of Polynomials...1 Multiplication of Polynomials... Multiplying a monomial by a monomial... Multiplying a monomial by a polynomial...
More informationMAIN IDEA The rectangle at the right has an area of 20 square units. The distance around the rectangle is 5 + 4 + 5 + 4, or 18 units.
19 Algebra: Area Formulas MAIN IDEA The rectangle at the right has an area of 20 square units. The distance around the rectangle is 5 + 4 + 5 + 4, or 1. Find the areas of rectangles and squares. New Vocabulary
More information2.4. Factoring Quadratic Expressions. Goal. Explore 2.4. Launch 2.4
2.4 Factoring Quadratic Epressions Goal Use the area model and Distributive Property to rewrite an epression that is in epanded form into an equivalent epression in factored form The area of a rectangle
More information10 7, 8. 2. 6x + 30x + 36 SOLUTION: 89 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
More informationNSM100 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
More informationChris Yuen. Algebra 1 Factoring. Early High School 810 Time Span: 5 instructional days
1 Chris Yuen Algebra 1 Factoring Early High School 810 Time Span: 5 instructional days Materials: Algebra Tiles and TI83 Plus Calculator. AMSCO Math A Chapter 18 Factoring. All mathematics material and
More informationPolynomial Functions
Polynomial Functions Lessons 71 and 73 Evaluate polynomial functions and solve polynomial equations. Lessons 7 and 79 Graph polynomial and square root functions. Lessons 74, 75, and 76 Find factors
More informationAnswer Key for California State Standards: Algebra I
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.
More informationName 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
More information85 Using the Distributive Property. Use the Distributive Property to factor each polynomial. 1. 21b 15a SOLUTION:
Use the Distributive Property to factor each polynomial. 1. 1b 15a The greatest common factor in each term is 3.. 14c + c The greatest common factor in each term is c. 3. 10g h + 9gh g h The greatest common
More informationMathematics Placement
Mathematics Placement The ACT COMPASS math test is a selfadaptive test, which potentially tests students within four different levels of math including prealgebra, algebra, college algebra, and trigonometry.
More informationNegative Integral Exponents. If x is nonzero, the reciprocal of x is written as 1 x. For example, the reciprocal of 23 is written as 2
4 (4) Chapter 4 Polynomials and Eponents P( r) 0 ( r) dollars. Which law of eponents can be used to simplify the last epression? Simplify it. P( r) 7. CD rollover. Ronnie invested P dollars in a year
More informationIn 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,
More informationSolving Quadratic Equations
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
More informationA Concrete Introduction. to the Abstract Concepts. of Integers and Algebra using Algebra Tiles
A Concrete Introduction to the Abstract Concepts of Integers and Algebra using Algebra Tiles Table of Contents Introduction... 1 page Integers 1: Introduction to Integers... 3 2: Working with Algebra Tiles...
More informationSection A3 Polynomials: Factoring APPLICATIONS. A22 Appendix A A BASIC ALGEBRA REVIEW
A Appendi A A BASIC ALGEBRA REVIEW C In Problems 53 56, perform the indicated operations and simplify. 53. ( ) 3 ( ) 3( ) 4 54. ( ) 3 ( ) 3( ) 7 55. 3{[ ( )] ( )( 3)} 56. {( 3)( ) [3 ( )]} 57. Show by
More informationStudents will be able to simplify and evaluate numerical and variable expressions using appropriate properties and order of operations.
Outcome 1: (Introduction to Algebra) Skills/Content 1. Simplify numerical expressions: a). Use order of operations b). Use exponents Students will be able to simplify and evaluate numerical and variable
More informationSystems of Equations Involving Circles and Lines
Name: Systems of Equations Involving Circles and Lines Date: In this lesson, we will be solving two new types of Systems of Equations. Systems of Equations Involving a Circle and a Line Solving a system
More informationFactor 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
More informationCOMPETENCY TEST SAMPLE TEST. A scientific, nongraphing calculator is required for this test. C = pd or. A = pr 2. A = 1 2 bh
BASIC MATHEMATICS COMPETENCY TEST SAMPLE TEST 2004 A scientific, nongraphing calculator is required for this test. The following formulas may be used on this test: Circumference of a circle: C = pd or
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More informationThis is Factoring and Solving by Factoring, chapter 6 from the book Beginning Algebra (index.html) (v. 1.0).
This is Factoring and Solving by Factoring, chapter 6 from the book Beginning Algebra (index.html) (v. 1.0). This book is licensed under a Creative Commons byncsa 3.0 (http://creativecommons.org/licenses/byncsa/
More informationA Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions
A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25
More informationCAHSEE on Target UC Davis, School and University Partnerships
UC Davis, School and University Partnerships CAHSEE on Target Mathematics Curriculum Published by The University of California, Davis, School/University Partnerships Program 006 Director Sarah R. Martinez,
More informationQuick 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
More informationWe start with the basic operations on polynomials, that is adding, subtracting, and multiplying.
R. Polnomials In this section we want to review all that we know about polnomials. We start with the basic operations on polnomials, that is adding, subtracting, and multipling. Recall, to add subtract
More informationAnchorage School District/Alaska Sr. High Math Performance Standards Algebra
Anchorage School District/Alaska Sr. High Math Performance Standards Algebra Algebra 1 2008 STANDARDS PERFORMANCE STANDARDS A1:1 Number Sense.1 Classify numbers as Real, Irrational, Rational, Integer,
More informationMATH 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
More informationHow do you compare numbers? On a number line, larger numbers are to the right and smaller numbers are to the left.
The verbal answers to all of the following questions should be memorized before completion of prealgebra. Answers that are not memorized will hinder your ability to succeed in algebra 1. Number Basics
More informationFlorida Math for College Readiness
Core Florida Math for College Readiness Florida Math for College Readiness provides a fourthyear math curriculum focused on developing the mastery of skills identified as critical to postsecondary readiness
More informationAlgebra 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
More informationCRLS Mathematics Department Algebra I Curriculum Map/Pacing Guide
Curriculum Map/Pacing Guide page 1 of 14 Quarter I start (CP & HN) 170 96 Unit 1: Number Sense and Operations 24 11 Totals Always Include 2 blocks for Review & Test Operating with Real Numbers: How are
More informationCAMI Education linked to CAPS: Mathematics
 1  TOPIC 1.1 Whole numbers _CAPS curriculum TERM 1 CONTENT Mental calculations Revise: Multiplication of whole numbers to at least 12 12 Ordering and comparing whole numbers Revise prime numbers to
More information