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6.1 The Greatest Common Factors and Factoring by Grouping A. Finding the Greatest common Factor (GCF) of a List of Integers or a list of terms Greatest common Factor (GCF)—is the largest common factor of the integers in the list. Steps: 1. Write each of the numbers as a product of prime number using exponent for repeated number. 2. Identify the common prime factors. 3. The product of all common prime factors found in step 2 is the greatest common factor. If there are no common prime factors, the greatest common factor is 1. 4. The GCF of a list of common variables raised to powers is the variable raised to the smallest exponent in the list. Example 1. Find the GCF of: 1. 45, 60 and 108 2. 12 x 8 , 9x 5 and − 30x 8 B. Factoring Out the Greatest Common Factor Example 2. Factor completely: 1. 6 a + 18 2. − 6x 2 − 36x 4 3. 15 x 7 − 20 x 6 + 10 x 5 + 25 x 3 4 2 1
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5. 5(y − 2 ) − x(y − 2 ) 6. 7 xy 3 ( p + q ) − ( p + q ) C. Factoring by Grouping Example 3. Factor completely: 1. 2 xy + 5 y 2 − 4 x − 10 y 2. 3xy − 4 + x − 12 y 3. 2 x − 2 + x 3 − 3x 2