The least number of times x appears as a factor is three. There will be three factors of x in the GCF. GCF x x x x3 In Example 1, notice that the power of 3 in the GCF is the smallest of the powers when comparing x5 and x3. 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. YOU TRY 1 Find the greatest common factor of y4 and y7. EXAMPLE 2 Find the greatest common factor for each group of terms. a) 30k4, 10k9, 50k6 b) 12a8b, 42a5b7 c) 63c5d3, 18c3, 27c2d2 Solution a) The GCF of the coeffi cients, 30, 10, and 50, is 10. The smallest exponent on k is 4, so k4 will be part of the GCF. The GCF of 30k4, 10k9, and 50k6 is 10k4. b) The GCF of the coeffi cients, 12 and 42, is 6. The smallest exponent on a is 5, so a5 will be part of the GCF. The smallest exponent on b is 1, so b will be part of the GCF. The GCF of 12a8b and 42a5b7 is 6a5b. c) The GCF of the coeffi cients is 9. The smallest exponent on c is 2, so c2 will be part of the GCF. There is no d in the term 18c3, so there will be no d in the GCF. The GCF of 63c5d3, 18c3, and 27c2d2 is 9c2. YOU TRY 2 Find the greatest common factor for each group of terms. a) 16p7, 8p5, 40p8 b) r6s5, 9r8s3, 12r4s4 2 Factoring vs. Multiplying Polynomials Earlier we said that to factor an integer is to write it as the product of two or more integers. To factor a polynomial is to write it as a product of two or more polynomials. Throughout this chapter we will study different factoring techniques. We will begin by discussing how to factor out the greatest common factor. Factoring a polynomial is the opposite of multiplying polynomials. Let’s see how these procedures are related. 358 CHAPTER 7 Factoring Polynomials www.mhhe.com/messersmith
messersmith_power_intermediate_algebra_1e_ch4_7_10
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