Read the explanations, follow the examples, take notes, and complete the You Trys. In Section 1.1, we discussed writing a number as the product of factors: 18 3 6 T T T Product Factor Factor To factor an integer is to write it as the product of two or more integers. Therefore, 18 can also be factored in other ways: 18 1 18 18 2 9 18 1 (18) 18 2 (9) 18 3 (6) 18 2 3 3 The last factorization, 2 3 3 or 2 32, is called the prime factorization of 18 since all of the factors are prime numbers. (See Section 1.1.) The factors of 18 are 1, 2, 3, 6, 9, 18, 1, 2, 3, 6, 9, and 18. We can also write the factors as 1, 2, 3, 6, 9, and 18. (Read 1 as “plus or minus 1.”) In this chapter, we will learn how to factor polynomials, a skill that is used in many ways throughout algebra. 1 Find the GCF of a Group of Monomials Definition The greatest common factor (GCF) of a group of two or more integers is the largest common factor of the numbers in the group. For example, if we want to fi nd the GCF of 18 and 24, we can list their positive factors. 18: 1, 2, 3, 6, 9, 18 24: 1, 2, 3, 4, 6, 8, 12, 24 The greatest common factor of 18 and 24 is 6. We can also use prime factors. We begin our study of factoring polynomials by discussing how to fi nd the greatest common factor of a group of monomials. Write the definition of GCF in your own words. EXAMPLE 1 In-Class Example 1 Find the greatest common factor of a10 and a6. Answer: a6 Find the greatest common factor of x4 and x6. Solution We can write x4 as 1 x4, and we can write x6 as x4 x2. The largest power of x that is a factor of both x4 and x6 is x4. Therefore, the GCF is x4. Notice that the power of 4 in the GCF is the smallest of the powers when comparing x4 and x6. This will always be true. Note The exponent on the variable in the GCF will be the smallest exponent appearing on the variable in the group of terms. www.mhhe.com/messersmith SECTION 7.1 The Greatest Common Factor and Factoring by Grouping 389
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