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Algebra I Chapter 8

Algebra I Chapter 8

Method 2 Use a factor

Method 2 Use a factor tree.Animationca.algebra1.com90 = 9 · 109 = 3 · 3, 10 = 2 · 5All of the factors in the last step are prime. Thus, the prime factorization of 90is 2 · 3 · 3 · 5 or 2 · 3 2 · 5.Usually the factors are ordered from the least prime factor to the greatest.Factoring a monomial is similar to factoring a whole number. A monomial isin factored form when it is expressed as the product of prime numbers andvariables, and no variable has an exponent greater than 1.EXAMPLEPrime Factorization of a MonomialFinding theGCF ofdistances willhelp you make a scalemodel of the solarsystem. Visitca.algebra1.com tocontinue work onyour project.Factor -12 a 2 b 3 completely.-12 a 2 b 3 = -1 · 12 a 2 b 3 Express -12 as -1 · 12= -1 · 2 · 6 · a · a · b · b · b 12 = 2 · 6, a 2 = a · a, and b 3 = b · b · b= -1 · 2 · 2 · 3 · a · a · b · b · b 6 = 2 · 3Thus, -12 a 2 b 3 in factored form is -1 · 2 · 2 · 3 · a · a · b · b · b.Factor each monomial completely.1A. 38r s 2 t 1B. -66p q 2Greatest Common Factor Two or more numbers may have some commonprime factors. Consider the prime factorization of 48 and 60.48 = 2 · 2 · 2 · 2 · 3 Factor each number.60 = 2 · 2 · 3 · 5 Circle the common prime factors.The common prime factors of 48 and 60 are 2, 2, and 3.The product of the common prime factors, 2 · 2 · 3 or 12, is called the greatestcommon factor of 48 and 60. The greatest common factor (GCF) is thegreatest number that is a factor of both original numbers. The GCF of two ormore monomials can be found in a similar way.Greatest Common Factor (GCF)• The GCF of two or more monomials is the product of their common factorswhen each monomial is written in factored form.• If two or more integers or monomials have a GCF of 1, then the integers ormonomials are said to be relatively prime.Extra Examples at ca.algebra1.comLesson 8-1 Monomials and Factoring 421

EXAMPLEFinding GCFAlternativeMethodYou can also find thegreatest commonfactor by listing thefactors of each numberand finding which ofthe common factors isthe greatest.Consider Example 2.15: 1 , 3, 5, 1516: 1 , 2, 4, 8, 16The only commonfactor, and thereforethe greatest commonfactor, is 1.GEOMETRY The areas of two rectangles are 15 square inches and 16square inches, respectively. The length and width of both figures arewhole numbers. If the rectangles have the same width, what is thegreatest possible value for their widths?Find the GCF of 15 and 16.15 = 3 · 5 Factor each number.16 = 2 · 2 · 2 · 2 There are no common prime factors.The GCF of 15 and 16 is 1, so 15 and 16 are relatively prime. The width ofthe rectangles is 1 inch.2. What is the greatest possible value for the widths if the rectanglesdescribed above have areas of 84 square inches and 70 square inches,respectively?EXAMPLEGCF of a Set of MonomialsFind the GCF of 36 x 2 y and 54x y2z.36 x 2 y = 2 · 2 · 3 · 3 · x · x · y Factor each number.54x y2z = 2 · 3 · 3 · 3 · x · y · y · zCircle the common prime factors.The GCF of 36 x 2 y and 54x y2z is 2 · 3 · 3 · x · y or 18xy.Find the GCF of each set of monomials.3A. 17 d 3 , 5 d 2 3B. 22 p 2 q, 32p r 2 tPersonal Tutor at ca.algebra1.comExample 1(p. 421)Factor each monomial completely.1. 4 p 2 2. 39b 3 c 2 3. -100 x 3 y z24. GARDENING Corey is planting 120 jalapeno pepper plants in a rectangulararrangement in his garden. In what ways can he arrange them so that hehas the same number of plants in each row, at least 4 rows of plants, and atleast 6 plants in each row?Examples 2, 3(p. 422)Find the GCF of each set of monomials.5. 10, 15 6. 54, 637. 18xy, 36 y 2 8. 25n, 21m9. 12qr, 8 r 2 , 16rs 10. 42 a 2 b, 6 a 2 , 18 a 3422 Chapter 8 Factoring

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