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Chapter 6 Section 1 6.1 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. Copyright © 2012, 2008, 2004 Pearson Education, Inc. The Greatest Common Factor: Factoring 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. For example, Multiplying Factoring 6 · 2 = 12 12 = 6 · 2 Factors Product Product other factored forms of 12 are − 6(−2), 3 · 4, −3(−4), 12 · 1, and Factors −12(−1). More than two factors may be used, so another factored form of 12 is 2 · 2 · 3. The positive integer factors of 12 are 1, 2, 3, 4, 6, 12. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 6.1-3 Objective 1 Find the greatest common factor of a list of terms. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 6.1-4 Find the greatest common factor of a list of terms. An integer that is a factor of two or more integers is called a common factor of those integers. For example, 6 is a common factor of 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. Recall from Chapter 1 that a prime number has only itself and 1 as factors. In Section 1.1, we factored numbers into prime factors. This is the first step in finding the greatest common factor of a list of numbers. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 6.1-5 Find the greatest common factor of a list of terms. (cont’d) Factors of a number are also divisors of the number. The greatest common factor is actually the same as the greatest common divisor. The are many rules for deciding what numbers to divide into a given number. Here are some especially useful divisibility rules for small numbers. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 6.1-6 Find the greatest common factor of a list of terms. (cont’d) Finding the Greatest Common Factor (GCF) Step 1: Factor. Write each number in prime factored form. Step 2: 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.) Step 3: Choose least exponents. Use as exponents on the common prime factors the least exponent from the prime factored forms. Step 4: Multiply. Multiply the primes from Step 3. If there are no primes left after Step 3, the greatest common factor is 1. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 6.1-7 EXAMPLE 1 Finding the Greatest Common Factor for Numbers Find the greatest common factor for each list of numbers. Solution: 50, 75 GCF = 25 12, 18, 26, 32 GCF = 2 50 2 5 5 75 3 5 5 12 2 2 3 26 2 13 18 2 3 3 32 2 2 2 2 2 22 2 11 24 2 2 2 3 22, 23, 24 GCF = 1 23 1 23 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 6.1-8 Find the greatest common factor of a list of terms. (cont’d) The GCF can also be found for a list of variable terms. The exponent on a variable in the GCF is the least exponent that appears in all the common factors. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 6.1-9 EXAMPLE 2 Finding the Greatest Common Factor for Variable Terms Find the greatest common factor for each list of terms. Solution: 16r 9 , 10r15 , 8r12 16r 9 1 2 2 2 2 r 9 GCF = 2r 9 10r15 1 2 5 r15 8r12 2 2 2 r12 s 4t 5 , s3t 6 , s9t 2 s 4t 5 s 4 t 5 GCF = s 3t 2 s 3t 6 s 3 t 6 s 9t 2 s 9 t 2 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 6.1-10 Objective 2 Factor out the greatest common factor. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 6.1-11 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 GCF of these terms is 3. We can write 3m + 12 so that each term is a product of 3 as one factor. 3m + 12 = 3 · m + 3 · 4 = 3(m + 4) GCF = 3 Distributive property The factored form of 3m + 12 is 3(m + 4). This process is called factoring out the greatest common factor. The polynomial 3m + 12 is not in factored form when written as 3 · m + 3 · 4. The terms are factored, but the polynomial is not. The factored form of 3m +12 is the product 3(m + 4). Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 6.1-12 EXAMPLE 3 Factoring Out the Greatest Common Factor Write in factored form by factoring out the greatest common factor. Solution: 6x 2 x 2 2 6 x 4 12 x 2 30t 25t 10t 6 5 r r 12 5t 4 6t 2 5t 2 4 r 10 10 r 2 1 8 p q 16 p q 12 p q 4 p q 2 p 4 p q 3q 5 2 6 3 4 7 4 2 2 5 Be sure to include the 1 in a problem like r12 + r10. Always check that the factored form can be multiplied out to give the original polynomial. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 6.1-13 EXAMPLE 4 Factoring Out the Greatest Common Factor Write in factored form by factoring out the greatest common factor. Solution: 6 p q r p q p q 6 r y 4 y 3 4 y 3 y 3 y 4 4 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 6.1-14 Objective 3 Factor by grouping. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 6.1-15 Factor by grouping. When a polynomial has four terms, common factors can sometimes be used to factor by grouping. Factoring a Polynomial with Four Terms by Grouping Step 1: Group terms. Collect the terms into two groups so that each group has a common factor. Step 2: Factor within groups. Factor out the greatest common factor from each group. Step 3: Factor the entire polynomial. Factor out a common binomial factor from the results of Step 2. Step 4: If necessary, rearrange terms. If Step 2 does not result in a common binomial factor, try a different grouping. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 6.1-16 EXAMPLE 5 Factoring by Grouping Factor by grouping. Solution: pq 5q 2 p 10 q p 5 2 p 5 p 5 q 2 2 xy 3 y 2 x 3 y 2 x 3 1 2 x 3 2 x 3 y 1 2a 4a 3ab 6b 2 2a a 2 3b a 2 a 2 2a 3b x3 3x 2 5 x 15 x2 x 3 5 x 3 Copyright © 2012, 2008, 2004 Pearson Education, Inc. x 3 x 2 5 Slide 6.1-17 EXAMPLE 6 Rearranging Terms before Factoring by Grouping Factor by grouping. Solution: 6 y 2 20w 15 y 8 yw 6 y 2 15 y 20w 8 yw 3 y 2 y 5 4w 2 y 5 2 y 5 3 y 4w 9mn 4 12m 3n 9mn 12m 3n 4 3m 3n 4 1 3n 4 3m 1 3n 4 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 6.1-18