1. Chapter 6 Section 1 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

2. The Greatest Common Factor; Factoring by Grouping 1 1 3 3 2 26.16.1 Find the greatest common factor of a list of terms. Factor out the greatest common factor. Factor by grouping.

3. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley other factored forms of 12 are − 6(−2), 3 · 4, −3(−4), 12 · 1, and −12(−1). Find the greatest common factor of a list of terms. Recall from Chapter 1 that to factor means “to write a quantity as a product.” For example, MultiplyingFactoring 6 · 2 = 12 12 = 6 · 2 Slide 6.1 - 3 Factors Product 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.

4. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 1 Objective 1 Slide 6.1 - 4 Find the greatest common factor of a list of terms.

5. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Find the greatest common factor of a list of terms. Slide 6.1 - 5 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.

6. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Find the greatest common factor of a list of terms. (cont’d) Slide 6.1 - 6 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.

7. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Find the greatest common factor of a list of terms. (cont’d) Slide 6.1 - 7 Find the greatest common factor (GCF) of a list of numbers as follows. Step 1: Factor. write each number in a prime factored form. Step 2: List common factors. List each prime number that is a factor of every number 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.

8. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Finding the Greatest Common Factor for Numbers Solution: Slide 6.1 - 8 Find the greatest common factor for each list of numbers. 50, 75 12, 18, 26, 32 12, 13, 14

9. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Finding the Greatest Common Factor for Variable Terms Solution: Slide 6.1 - 9 Find the greatest common factor for each list of terms.

10. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2 Objective 2 Factor out the greatest common factor. Slide 6.1 - 10

11. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Factor out the greatest common factor. Slide 6.1 - 11 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. The polynomial 3 m + 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 3 m +12 is the product 3( m + 4). 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. Distributive Property.

12. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Solution: Factoring Out the Greatest Common Factor Slide 6.1 - 12 Factor out the GCF. In the fifth example, use fractions in the factored form. Be sure to include the 1 in a problem like r 12 + r 10. Always check that the factored form can be multiplied out to give the original polynomial.

13. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Solution: Factoring Out the Greatest Common Factor Slide 6.1 - 13 Factor out the greatest common factor.

14. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3 Objective 3 Factor by grouping. Slide 6.1 - 14

15. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Factor by grouping. Slide 6.1 - 15 When a polynomial has four terms, common factors can sometimes be used to factor 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.

16. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Factoring by Grouping Slide 6.1 - 16 Factor by grouping. Solution:

17. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Rearranging Terms before Factoring by Grouping Slide 6.1 - 17 Factor by grouping. Solution: