Note that the words divisor and factor are equivalent. They have the same meaning.

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1 2 MODULE 5. FACTORING 5a Common Factors The Greatest Common Factor We begin this section with definitions of factors and divisors. Because 24 = 2 12, both 2 and 12 are factors of 24. However, note that 2 is also a divisor of 24, because when you divide 24 by 2 you get 12, with a remainder of zero. Similarly, 12 is also a divisor of 24, because when you divide 24 by 12 you get 2, with a remainder of zero. Factors and divisors. Suppose m and n are integers. Then m is a divisor (factor) ofn if and only if there exists another integer k so that n = m k. Note that the words divisor and factor are equivalent. They have the same meaning. EXAMPLE 1. List the positive divisors (factors) of 24. Solution: First, list all possible ways that we can express 24 as a product of two positive integers: 24 = 1 24 or 24 = 2 12 or 24 = 3 8 or 24 = 4 6 Therefore, the positive divisors (factors) of 24 are 1, 2, 3, 4, 6, 8, and 24. EXAMPLE 2. common. List the positive divisors (factors) that 36 and 48 have in Solution: First, list all positive divisors (factors) of 36 and 48 separately, then box the divisors that are in common. Divisors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, 36 Divisors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 Therefore, the common positive divisors (factors) of 36 and 48 are 1, 2, 3, 4, 6, and 12.

2 5A. COMMON FACTORS 3 Greatest common divisor. The greatest common divisor (factor) of a and b is the largest positive number that divides evenly (no remainder) both a and b. The greatest common divisor of a and b is denoted by the symbolism gcd(a, b). We will also use the abbreviation gcf(a, b) to represents the greatest common factor of a and b. Remember, greatest common divisor and greatest common factor have the same meaning. In Example 2, we listed the common positive divisors of 36 and 48. The largest of these common divisors was 12. Hence, the greatest common divisor (factor) of 36 and 48 is 12, written gcd(36, 48) = 12. With smaller numbers, it is usually easy to identify the greatest common divisor (factor). EXAMPLE 3. State the greatest common divisor (factor) of each of the following pairs of numbers: (a) 18 and 24, (b) 30 and 40, and (c) 16 and 24. Solution: In each case, we must find the largest possible positive integer that divides evenly into both the given numbers. a) The largest positive integer that divides evenly into both 18 and 24 is 6. Thus, gcd(18, 24) = 6. b) The largest positive integer that divides evenly into both 30 and 40 is 10. Thus, gcd(30, 40) = 10. c) The largest positive integer that divides evenly into both 16 and 24 is 8. Thus, gcd(16, 24) = 6. With larger numbers, it is harder to identify the greatest common divisor (factor). However, prime factorization will save the day! EXAMPLE 4. Find the greatest common divisor (factor) of 360 and 756. Solution: Prime factor 360 and 756, writing your answer in exponential form

3 4 MODULE 5. FACTORING Thus: 360 = = To find the greatest common divisor (factor), list each factor that appears in common to the highest power that appears in common. In this case, the factors 2 and 3 appear in common, with 2 2 being the highest power of 2 and 3 2 being the highest power of 3 that appear in common. Therefore, the greatest common divisor of 360 and 756 is: GCD = =4 9 =36 Therefore, the greatest common divisor (factor) is gcd(360, 756) = 36. Note what happens when we write each of the given numbers as a product of the greatest common factor and a second factor: 360 = = In each case, note how the second second factors (10 and 21) contain no additional common factors. Finding the Greatest Common Factor of Monomials Example 4 reveals the technique used to find the greatest common factor of two or more monomials. Finding the GCF of two or more monomials. To find the greatest common factor of two or more monomials, proceed as follows: 1. Find the greatest common factor (divisor) of the coefficients of the given monomials. Use prime factorization if necessary. 2. List each variable that appears in common in the given monomials. 3. Raise each variable that appears in common to the highest power that appears in common among the given monomials.

4 5A. COMMON FACTORS 5 ou Try It! EXAMPLE 5. Find the greatest common factor of 6x 3 y 3 and 9x 2 y 5. Solution: To find the GCF of 6x 3 y 3 and 9x 2 y 5,wenotethat: 1. The greatest common factor (divisor) of 6 and 9 is The monomials 6x 3 y 3 and 9x 2 y 5 have the variables x and y in common. 3. The highest power of x in common is x 2. The highest power of y in common is y 3. Thus, the greatest common factor is gcf(6x 3 y 3, 9x 2 y 5 )=3x 2 y 3. Note what happens when we write each of the given monomials as a product of the greatest common factor and a second monomial: 6x 3 y 3 = 3x 2 y 3 2x 9x 2 y 5 = 3x 2 y 3 3y Note how the set of second monomial factors (2x and 3y) contain no additional common factors. EXAMPLE 6. Find the greatest common factor of 12x 4,18x 3,and30x 2. Solution: To find the GCF of 12x 4,18x 3,and30x 2,wenotethat: 1. The greatest common factor (divisor) of 12, 18, and 30 is The monomials 12x 4,18x 3,and30x 2 have the variable x in common. 3. The highest power of x in common is x 2. Thus, the greatest common factor is gcf(12x 4, 18x 3, 30x 2 )=6x 2. Note what happens when we write each of the given monomials as a product of the greatest common factor and a second monomial: 12x 4 = 6x 2 2x 2 18x 3 = 6x 2 3x 30x 2 = 6x 2 5 Note how the set of second monomial factors (2x 2, 3x, and 5) contain no additional common factors.

5 6 MODULE 5. FACTORING Factor Out the GCF Previously we multiplied a monomial and polynomial by distributing the monomial times each term in the polynomial. 2x(3x 2 +4x 7) = 2x 3x 2 + 2x 4x 2x 7 =6x 3 +8x 2 14x In this section we reverse that multiplication process. THat is, you are given the final product and are asked to find original multiplication problem. In the case 6x 3 +8x 2 14x, thegreatestcommonfactorof6x 3,8x 2,and14x is 2x. We then use the distributive property to factor out 2x from each term of the polynomial. 6x 3 +8x 2 14x = 2x 3x 2 + 2x 4x 2x 7 = 2x(3x 2 +4x 7) Factoring. Factoring is unmultiplying. You are given the product, then asked to find the original multiplication problem. First rule of factoring. If the terms of the given polynomial have a greatest common factor (GCF), then factor out the GCF. EXAMPLE 7. Factor: 6x 2 +10x +14 Solution: The greatest common factor (GCF) of 6x 2,10x and 14 is 2. Factor out the GCF. 6x 2 +10x +14=2 3x x = 2(3x 2 +5x +7) Checking your work. check your work. Every time you factor a polynomial, remultiply to Check: Multiply. Distribute the 2. 2(3x 2 +5x +7) = 2 3x x =6x 2 +10x +14 That s the original polynomial, so we factored correctly.

6 5A. COMMON FACTORS 7 ou Try It! EXAMPLE 8. Factor: 12y 5 32y 4 +8y 2 Solution: The greatest common factor (GCF) of 12y 5,32y 4 and 8y 2 is 4y 2. Factor out the GCF. 12y 5 32y 4 +8y 2 = 4y 2 3y 3 4y 2 8y 2 + 4y 2 2 = 4y 2 (3y 3 8y 2 +2) Check: Multiply. Distribute the monomial 4y 2. 4y 2 (3y 3 8y 2 +2) = 4y 2 3y 3 4y 2 8y 2 + 4y 2 2 =12y 5 32y 4 +8y 2 That s the original polynomial. We have factored correctly. EXAMPLE 9. Factor: 12a 3 b +24a 2 b 2 +12ab 3 Solution: The greatest common factor (GCF) of 12a 3 b,24a 2 b 2 and 12ab 3 is 12ab. FactorouttheGCF. 12a 3 b +24a 2 b 2 +12ab 3 = 12ab a 2 12ab 2ab + 12ab b 2 = 12ab(a 2 +2ab + b 2 ) Check: Multiply. Distribute the monomial 12ab. 12ab(a 2 +2ab + b 2 ) = 12ab a 2 12ab 2ab + 12ab b 2 =12a 3 b +24a 2 b 2 +12ab 3 That s the original polynomial. We have factored correctly. Speeding Things Up a Bit After showing your work on a number of examples such as those in Examples 7, 8, and 9, you ll need to learn how to perform the process mentally. EXAMPLE 10. Factor each of the following polynomials: (a) 24x + 32, (b) 5x 3 10x 2 10x, and(c)2x 4 y +2x 3 y 2 6x 2 y 3. Solution: In each case, factor out the greatest common factor (GCF):

7 8 MODULE 5. FACTORING a) The GCF of 24x and 32 is 8. Thus, 24x +32=8(3x +4) b) The GCF of 5x 3,10x 2,and10x is 5x. Thus: 5x 3 10x 2 10x =5x(x 2 2x 2) c) The GCF of 2x 4 y,2x 3 y 2,and6x 2 y 3 is 2x 2 y.thus: 2x 4 y +2x 3 y 2 6x 2 y 3 =2x 2 y(x 2 + xy 3y 2 ) When mentally factoring out the GCF, it is even more important that you check your results. The check can also be done mentally. For example, in checking the third result, mentally distribute 2x 2 y times each term of x 2 + xy 3y 2. For example, multiplying 2x 2 y times the first term x 2 produces 2x 4 y,thefirst term in the original polynomial. 2x 2 y (x 2 + xy 3y 2 )=2x 4 y +2x 3 y 2 6x 2 y 3 Continue in this manner by checking the product of 2x 2 y with each term of x 2 + xy 3y 2. Make sure that each result agrees with the corresponding term of the original polynomial. The distributive property allows us to pull the GCF out in front or to pull it out in back. In symbols: ab + ac = a(b + c) or ba + ca =(b + c)a EXAMPLE 11. Factor: 2x(3x +2)+5(3x +2) Solution: In this case, the greatest common factor (GCF) is 3x +2. 2x(3x +2)+5(3x +2)=2x (3x +2)+5 (3x +2) =(2x +5)(3x +2) EXAMPLE 12. Factor: 15a(a + b) 12(a + b) Solution: In this case, the greatest common factor (GCF) is 3(a + b). 15a(a + b) 12(a + b) =3(a + b) 5a 3(a + b) 4 = 3(a + b)(5a 4)

8 5A. COMMON FACTORS 9 Alternate solution: It is possible that you might fail to notice that 15 and 12 are divisible by 3, factoring out only a common factor a + b. 15a(a + b) 12(a + b) =15a (a + b) 12 (a + b) =(15a 12)(a + b) However, you now need to notice that you can continue, factoring out a 3 from both 15a and 12. =3(5a 4)(a + b) Note that the order of factors differs from the first solution, but because of the commutative property of multiplication, the order does not matter. The answers are the same. Factoring by Grouping The final factoring skill in this section involves four-term expressions. The technique for factoring a four-term expression is called factoring by grouping. EXAMPLE 13. Factor: 2x 2 +3x +2xy +3y Solution: We group the first and second terms, noting that we can factor an x out of both of these terms. Then we group the third and fourth terms, noting that we can factor a y out of both of these terms. 2x 2 +3x +2xy +3y = x (2x +3)+ y (2x +3) Now we note that we can factor a 2x +3outofbothoftheseterms. =(x + y)(2x +3) EXAMPLE 14. Factor: ac bd + ad bc Solution: In the current order, there really isn t any common factor in the first and second terms that we can factor out, nor is there a common factor in the third and fourth terms that we can factor out. ac bd + ad bc

9 10 MODULE 5. FACTORING Upon rearranging the terms, we note that we can factor out an a from the first and second terms and a b from the third and fourth terms. ac + ad bd bc = a (c + d) + b ( d c) However, this is not helpful, because it does not produce a common factor. Let s try factoring a b from the third and fourth terms instead. ac + ad bd bc = a (c + d) b (d + c) At first glance, this still doesn t seem helpful until we realize that d + c is the same as c + d. = a (c + d) b (c + d) Now we can factor a c + d out of both of these terms. =(a b)(c + d) EXAMPLE 15. Factor: 2xy 2xz + yz y 2 Solution: Note that we can factor a 2x out of the first two terms and a y out of the second two terms. 2xy 2xz + yz y 2 =2x (y z) + y (z y) However, this is not helpful, because it does not produce a common factor. Let s try factoring a y from the third and fourth terms instead. 2xy 2xz + yz y 2 =2x (y z) y (y z) That produced a common factor of y z, which we can now factor out. =(2x + y)(y z)