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## 5.1 The Greatest Common Factor; Factoring by Grouping

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**Product**Factors The Greatest Common Factor: Factoring by Grouping Recall from Section 1.1 that to factormeans “to write a quantity as a product.” That is, factoring is the opposite of multiplying. For example, Multiplying Factoring 6 · 2 = 1212 = 6 · 2 Factors Product other factored formsof 12 are − 6(−2), 3 · 4, −3(−4), 12 · 1, and −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. Slide 5.1-3**Objective 1**Find the greatest common factor of a list of terms. Slide 5.1-4**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. Slide 5.1-5**Find the greatest common factor of a list of terms.**(cont’d) Factorsof a number are also divisorsof the number. The greatest common factoris actually the same as the greatest common divisor. There are many rules for deciding what numbers to divide into a given number. Here are some especially useful divisibility rules for small numbers. Slide 5.1-6**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 theleastexponent from the prime factored forms. Step 4:Multiply.Multiply the primes fromStep 3.If there are no primes left afterStep 3,the greatest common factor is 1. Slide 5.1-7**CLASSROOM EXAMPLE 1**Finding the Greatest Common Factor for Numbers Solution: Find the greatest common factor for each list of numbers. 50, 75 12, 18, 26, 32 22, 23, 24 Slide 5.1-8**Find the greatest common factor of a list of terms.**(cont’d) Find the greatest common factor of a list of terms. 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. Slide 5.1-9**CLASSROOM EXAMPLE 2**Finding the Greatest Common Factor for Variable Terms Solution: Find the greatest common factor for each list of terms. Slide 5.1-10**Objective 2**Factor out the greatest common factor. Slide 5.1-11**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) The factored form of 3m + 12 is 3(m + 4). This process is called factoring out the greatest common factor. GCF = 3 Distributive property The polynomial 3m + 12 is notin factored form when written as 3 · m+ 3 · 4. The terms arefactored, but the polynomial is not. The factored form of 3m +12 is the product 3(m + 4). Slide 5.1-12**CLASSROOM EXAMPLE 3**Factoring Out the Greatest Common Factor Write in factored form by factoring out the greatest common factor. Solution: 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. Slide 5.1-13**CLASSROOM EXAMPLE 4**Factoring Out the Greatest Common Factor Write in factored form by factoring out the greatest common factor. Solution: Slide 5.1-14**Objective 3**Factor by grouping. Slide 5.1-15**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 ofStep 2. Step 4:If necessary, rearrange terms.If Step 2does not result in a common binomial factor, try a different grouping. Slide 5.1-16**CLASSROOM EXAMPLE 5**Factoring by Grouping Solution: Factor by grouping. Slide 5.1-17**CLASSROOM EXAMPLE 6**Rearranging Terms before Factoring by Grouping Solution: Factor by grouping. Slide 5.1-18