The Greatest Common Factor; Factoring by Grouping


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1 296 CHAPTER 5 Factoring and Applications 5.1 The Greatest Common Factor; Factoring by Grouping OBJECTIVES 1 Find the greatest common factor of a list of terms. 2 Factor out the greatest common factor. 3 Factor by grouping. Recall from Section 1.1 that to factor means to write a quantity as a product. That is, factoring is the opposite of multiplying. Multiplying Other factored forms of 12 are , 6 # 2 = 12 Factoring Factors Product Product Factors 3 # 4, , 12 # 1, 12 = 6 # 2 More than two factors may be used, so another factored form of 12 is 2 # 2 # 3. and OBJECTIVE 1 Find the greatest common factor of a list of terms. An integer that is a factor of two or more integers is a common factor of those integers. For example, 6 is a common factor of 18 and 24, since 6 is a factor of both 18 and 24. Other common factors of 18 and 24 are 1, 2, and 3. The greatest common factor (GCF) of a list of integers is the largest common factor of those integers. Thus, 6 is the greatest common factor of 18 and 24, since it is the largest of their common factors. NOTE Factors of a number are also divisors of the number. The greatest common factor is actually the same as the greatest common divisor. Here are some useful divisibility rules for deciding what numbers divide into a given number. A Whole Number Divisible by 2 Ends in 0, 2, 4, 6, or 8 Must Have the Following Property: 3 Sum of digits divisible by 3 4 Last two digits form a number divisible by 4 5 Ends in 0 or 5 6 Divisible by both 2 and 3 8 Last three digits form a number divisible by 8 9 Sum of digits divisible by 9 10 Ends in 0 Finding the Greatest Common Factor (GCF) Step 1 Step 2 Step 3 Step 4 Factor. Write each number in prime factored form. List common factors. List each prime number or each variable that is a factor of every term in the list. (If a prime does not appear in one of the prime factored forms, it cannot appear in the greatest common factor.) Choose least exponents. Use as exponents on the common prime factors the least exponents from the prime factored forms. Multiply the primes from Step 3. If there are no primes left after Step 3, the greatest common factor is 1.
2 SECTION 5.1 The Greatest Common Factor; Factoring by Grouping 297 EXERCISE 1 Find the greatest common factor for each list of numbers. 24, 36 54, 90, 108 (c) 15, 19, 25 EXAMPLE 1 Finding the Greatest Common Factor for Numbers Find the greatest common factor for each list of numbers. 30, = 2 # 3 # 5 45 = 3 # 3 # 5 Write the prime factored form of each number. Use each prime the least number of times it appears in all the factored forms. There is no 2 in the prime factored form of 45, so there will be no 2 in the greatest common factor. The least number of times 3 appears in all the factored forms is 1, and the least number of times 5 appears is also 1. 72, 120, = 2 # 2 # 2 # 3 # = 2 # 2 # 2 # 3 # 5 GCF = 3 1 # 5 1 = = 2 # 2 # 2 # 2 # 3 # 3 # 3 Write the prime factored form of each number. The least number of times 2 appears in all the factored forms is 3, and the least number of times 3 appears is 1. There is no 5 in the prime factored form of either 72 or 432. (c) 10, 11, = 2 # 5 11 = = 2 # 7 GCF = 2 3 # 3 1 = 24 Write the prime factored form of each number. There are no primes common to all three numbers, so the GCF is 1. The greatest common factor can also be found for a list of variable terms. For example, the terms x 4, x 5, x 6, and x 7 have x 4 as the greatest common factor because each of these terms can be written with x 4 as a factor. x 4 = 1 # x 4, x 5 = x # x 4, x 6 = x 2 # x 4, x 7 = x 3 # x 4 NOTE The exponent on a variable in the GCF is the least exponent that appears in all the common factors. ANSWERS (c) 1 EXAMPLE 2 Finding the Greatest Common Factor for Variable Terms Find the greatest common factor for each list of terms. 21m 7, 18m 6, 45m 8, 24m 5 21m 7 = 3 # 7 # m 7 18m 6 = 2 # 3 # 3 # m 6 45m 8 = 3 # 3 # 5 # m 8 24m 5 = 2 # 2 # 2 # 3 # m 5 Here, 3 is the greatest common factor of the coefficients 21, 18, 45, and 24. The least exponent on m is 5. GCF = 3m 5
3 298 CHAPTER 5 Factoring and Applications EXERCISE 2 Find the greatest common factor for each list of terms. 25k 3, 15k 2, 35k 5 m 3 n 5, m 4 n 4, m 5 n 2 x 4 y 2, x 7 y 5, x 3 x 4 y 2 = x 4 # y 7, y 15 x 7 y 5 = x 7 # y 2 There is no x in the last term, y 15, so x will not appear in x 3 y 7 = x 3 # y 5 the greatest common factor. There is a y in each term, however, and 2 is the least exponent on y. y 7 GCF = y 2 y 15 = y 15 OBJECTIVE 2 Factor out the greatest common factor. Writing a polynomial (a sum) in factored form as a product is called factoring. For example, the polynomial 3m + 12 has two terms: 3m and 12. The greatest common factor of these two terms is 3. We can write 3m + 12 so that each term is a product with 3 as one factor. 3m + 12 = 3 # m + 3 # 4 GCF = 3 = 31m + 42 Distributive property The factored form of 3m + 12 is 31m This process is called factoring out the greatest common factor. CAUTION The polynomial 3m + 12 is not in factored form when written as 3 # m + 3 # 4. Not in factored form The terms are factored, but the polynomial is not. The factored form of 3m + 12 is the product 31m In factored form EXAMPLE 3 Factoring Out the Greatest Common Factor Write in factored form by factoring out the greatest common factor. 5y y = 5y1 y2 + 5y122 GCF = 5y = 5y1 y + 22 Distributive property CHECK Multiply the factored form. ANSWERS 2. 5k 2 m 3 n 2 5y1 y + 22 = 5y1 y2 + 5y122 Distributive property = 5y y Original polynomial 20m m m 3 = 5m 3 14m m 3 12m2 + 5m GCF = 5m 3 = 5m 3 14m 2 + 2m + 32 Factor out 5m 3. CHECK 5m 3 14m 2 + 2m + 32 = 20m m m 3 Original polynomial
4 SECTION 5.1 The Greatest Common Factor; Factoring by Grouping 299 EXERCISE 3 Write in factored form by factoring out the greatest common factor. 7t 414t 3 8x 620x x 4 (c) 30m 4 n 342m 2 n 2 (c) x 5 + x 3 = x 3 1x x = x 3 1x Check mentally by distributing (d) 20m 7 p 236m 3 p 4 = 4m 3 p 2 15m 49p 2 2 GCF = x 3 Don t forget the 1. x 3 = 4m 3 p 2 15m 4 24m 3 p 2 19p 2 2 over each term inside the parentheses. GCF = 4m 3 p 2 Factor out 4m 3 p 2. CAUTION Be sure to include the 1 in a problem like Example 3(c). Check that the factored form can be multiplied out to give the original polynomial. EXERCISE 4 Write in factored form by factoring out the greatest common factor. x1x x + 22 a1t b1t EXAMPLE 4 Factoring Out the Greatest Common Factor Write in factored form by factoring out the greatest common factor. Same a1a a + 32 The binomial a + 3 is the greatest common factor. =1a + 321a + 42 x 2 1x x + 12 =1x + 121x 252 Factor out a + 3. Factor out x + 1. NOTE In factored forms like those in Example 4, the order of the factors does not matter because of the commutative property of multiplication. 1a + 321a + 42 can also be written 1a + 421a OBJECTIVE 3 Factor by grouping. When a polynomial has four terms, common factors can sometimes be used to factor by grouping. ANSWERS 3. 7t 3 1t x 4 12x 25x + 72 (c) 6m 2 n 2 15m 2 n x + 221x t a  b2 EXAMPLE 5 Factor by grouping. Factoring by Grouping 2x ax + 3a Group the first two terms and the last two terms, since the first two terms have a common factor of 2 and the last two terms have a common factor of a. 2x ax + 3a =12x ax + 3a2 = 21x a1x + 32 Factor each group. The expression is still not in factored form because it is the sum of two terms. Now, however, x + 3 is a common factor and can be factored out a21x + 32 is also correct. = 21x a1x + 32 =1x a2 x + 3 is a common factor. Factor out x + 3.
5 300 CHAPTER 5 Factoring and Applications EXERCISE 5 Factor by grouping. ab + 3a + 5b xy + 3x + 4y + 1 (c) x 3 + 5x 28x  40 The final result1x a2 is in factored form because it is a product. CHECK CHECK (c) CHECK 1x a2 = 2x + ax a = 2x ax + 3a 6ax + 24x + a + 4 =16ax + 24x2 +1a + 42 = 6x1a a + 42 =1a x a x + 12 = 6ax + a + 24x + 4 = 6ax + 24x + a + 4 2x 210x + 3xy  15y =12x 210x2 +13xy  15y2 = 2x1x y1x  52 =1x x + 3y2 1x x + 3y2 FOIL (Section 5.5) Rearrange terms to obtain the original polynomial. Factor each group. Remember the 1. Factor out a + 4. = 2x 2 + 3xy  10x  15y = 2x 210x + 3xy  15y FOIL Rearrange terms to obtain the original polynomial. Factor each group. Factor out x  5. FOIL Original polynomial (d) t 3 + 2t 23t  6 =1t 3 + 2t t  62 = t 2 1t t + 22 =1t + 221t 2 32 Check by multiplying. Be careful with signs. Write a + sign between the groups. Factor out 3 so there is a common factor, t + 2; 3(t + 2) = 3t  6. Factor out t + 2. CAUTION Be careful with signs when grouping in a problem like Example 5(d). It is wise to check the factoring in the second step, as shown in the side comment in that example, before continuing. ANSWERS 5. 1b + 321a y x + 12 (c) 1x + 521x 282 Factoring a Polynomial with Four Terms by Grouping Step 1 Step 2 Step 3 Step 4 Group terms. Collect the terms into two groups so that each group has a common factor. Factor within groups. Factor out the greatest common factor from each group. Factor the entire polynomial. Factor out a common binomial factor from the results of Step 2. If necessary, rearrange terms. If Step 2 does not result in a common binomial factor, try a different grouping.
6 SECTION 5.1 The Greatest Common Factor; Factoring by Grouping 301 EXERCISE 6 Factor by grouping. 12p 228q  16pq + 21p 5xy x + 2y ANSWERS 6. 13p  4q214p x y  32 EXAMPLE 6 Factor by grouping. Rearranging Terms before Factoring by Grouping 10x 212y + 15x  8xy Factoring out the common factor of 2 from the first two terms and the common factor of x from the last two terms gives the following. CHECK This does not lead to a common factor, so we try rearranging the terms. 10x 212y + 15x  8xy = 10x 28xy  12y + 15x =110x 28xy y + 15x2 = 2x15x  4y y + 5x2 = 2x15x  4y x  4y2 =15x  4y212x x  4y212x x 212y + 15x  8xy = 10x x  8xy  12y = 10x 212y + 15x  8xy Commutative property Factor each group. Rewrite 4y + 5x. Factor out 5x  4y. FOIL Original polynomial 2xy y  8x We need to rearrange these terms to get two groups that each have a common factor. Trial and error suggests the following grouping. 2xy y  8x =12xy  3y2 +18x = y12x x  32 =12x y  42 = 215x 26y2 + x1158y2 Write a + sign between the groups. Be careful with signs. Factor each group; 412x  32 = 8x Factor out 2x  3. Since the quantities in parentheses in the second step must be the same, we factored out 4 rather than 4. Check by multiplying. 5.1 EXERCISES Complete solution available on the Video Resources on DVD Find the greatest common factor for each list of numbers. See Example , 20, , 30, , 24, 36, , 30, 45, , 8, , 22, 23 Find the greatest common factor for each list of terms. See Examples 1 and y, w, x 3, 40x 6, 50x z 4, 70z 8, 90z x 4 y 3, xy a 4 b 5, a 3 b m 3 n 2, 18m 5 n 4, 36m 8 n p 5 r 7, 30p 7 r 8, 50p 5 r 3
7 302 CHAPTER 5 Factoring and Applications Concept Check An expression is factored when it is written as a product, not a sum. Which of the following are not factored? 15. 2k 2 15k k 2 15k k 2 +15k k 2 + 5k2 + 1 Complete each factoring by writing each polynomial as the product of two factors m p = 3m = 6p z 9 = 4z k m 4 n a 3 b 2 = 5k = 3m 3 n1 2 = 9a 2 b y p a 220a = = = 10a x 230x 29. 8x 2 y + 12x 3 y = 15x1 2 = 4x 2 y1 2 18s 3 t st = 2st How can you check your answer when you factor a polynomial? 32. Concept Check A student factored 18x 3 y 2 + 9xy as 9xy12x 2 y2. WHAT WENT WRONG? Factor correctly. Write in factored form by factoring out the greatest common factor. See Examples 3 and x 24x 34. m 27m 35. 6t t 36. 8x 2 + 6x m 39m p 324p z z k4 + 15k x 3 + 6x b 3 + 7b y y a a w z mn m 2 n p 2 y + 38p 2 y y y 439y x5 + 25x420x p 6 q + 45p 5 q p 3 q a3z5 + 60a4z4 + 85a5z2 53. a 5 + 2a 3 b 23a 5 b 2 + 4a 4 b x 6 + 5x 4 y 36xy xy 55. c1x d1x r1x t1x m1m + 2n2 + n1m + 2n2 58. q1q + 4p2 + p1q + 4p2 59. q 2 1 p p y 2 1x x  92 Students often have difficulty when factoring by grouping because they are not able to tell when the polynomial is completely factored. For example, 5y12x t12x  32 Not in factored form is not in factored form, because it is the sum of two terms: 5y12x  32 and 8t12x However, because 2x  3 is a common factor of these two terms, the expression can now be factored. 12x y + 8t2 In factored form The factored form is a product of two factors: 2x  3 and 5y + 8t. Concept Check Determine whether each expression is in factored form or is not in factored form. If it is not in factored form, factor it if possible t x17t r15x x x217t r x x 2 1 y y k 3 1s s Concept Check Why is it not possible to factor the expression in Exercise 65?
8 SECTION 5.1 The Greatest Common Factor; Factoring by Grouping Concept Check A student factored x 3 + 4x 22x  8 as follows. x 3 + 4x 22x  8 The student could not find a common factor of the two terms. WHAT WENT WRONG? Complete the factoring. Factor by grouping. See Examples 5 and 6. =1x 3 + 4x x  82 = x 2 1x x p 2 + 4p + pq + 4q 70. m 2 + 2m + mn + 2n 71. a 22a + ab  2b 72. y 26y + yw  6w 73. 7z z  az  2a 74. 5m mp  2mr  6pr r ry  3xr  2xy 76. 8s24st + 6sy  3yt 77. 3a 3 + 3ab 2 + 2a 2 b + 2b x 3 + 3x 2 y + 4xy 2 + 3y a  3b + ab x  2y + xy m 34m 2 p 24mp + p t 32t 2 s 25ts + s y 2 + 3x + 3y + xy 84. m p + 7m + 2mp 85. 5m  6p  2mp y  9x  3xy r 22ty + 12ry  3rt a 24bc + 16ac  3ab 89. a a 5 b  6b 90. b ab 310a RELATING CONCEPTS EXERCISES FOR INDIVIDUAL OR GROUP WORK In many cases, the choice of which pairs of terms to group when factoring by grouping can be made in different ways. To see this for Example 6, work Exercises in order. 91. Start with the polynomial from Example 6, 2xy y  8x, and rearrange the terms as follows: 2xy  8x  3y What property from Section 1.7 allows this? 92. Group the first two terms and the last two terms of the rearranged polynomial in Exercise 91. Then factor each group. 93. Is your result from Exercise 92 in factored form? Explain your answer. 94. If your answer to Exercise 93 is no, factor the polynomial. Is the result the same as that shown for Example 6? PREVIEW EXERCISES Find each product. See Section x + 621x x  321x x + 221x x1x + 521x x 2 1x 2 + 3x x 2 12x 24x  92
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