Chapter 7: Summary Definition/Procedure Example 7.1 The Greatest Common Factor and Factoring by Grouping To factor a polynomial is to write it as a product of two or more polynomials. To factor out a greatest common factor (GCF): 1) Identify the GCF of all of the terms of the polynomial. 2) Rewrite each term as the product of the GCF and another factor. 3) Use the distributive property to factor out the GCF from the terms of the polynomial. 4) Check the answer by multiplying the factors. (p. 391) The fi rst step in factoring any polynomial is to ask yourself, “Can I factor out a GCF?” The last step in factoring any polynomial is to ask yourself, “Can I factor again?” Try to factor by grouping when you are asked to factor a polynomial containing four terms. 1) Make two groups of two terms so that each group has a common factor. 2) Take out the common factor from each group of terms. 3) Factor out the common factor using the distributive property. 4) Check the answer by multiplying the factors. (p. 394) Factor out the greatest common factor. 6k6 27k5 15k4 The GCF is 3k4. 6k6 27k5 15k4 (3k4)(2k2) (3k4)(9k) (3k4)(5) 3k4(2k2 9k 5) Check: 3k4(2k2 9k 5) 6k6 27k5 15k4 ✓ Factor 10xy 5y 8x 4 completely. Since the four terms have a GCF of 1, we will not factor out a GCF. Begin by grouping two terms together so that each group has a common factor. 10xy 5y 8x 4 5y(2x 1) 4(2x 1) Take out the common factor. (2x 1)(5y 4) Factor out (2x 1). Check: (2x 1)(5y 4) 10xy 5y 8x 4 ✓ 7.2 Factoring Trinomials of the Form x2 bx c Factoring x2 bx c If x2 bx c (x m)(x n), then 1) if b and c are positive, then both m and n must be positive. 2) if c is positive and b is negative, then both m and n must be negative. 3) if c is negative, then one integer, m, must be positive and the other integer, n, must be negative. (p. 400) 7.3 Factoring Trinomials of the Form ax2 bx c (a 1) Factoring ax2 bx c by grouping. (p. 406) Factor completely. a) t2 9t 20 Think of two numbers whose product is 20 and whose sum is 9: 4 and 5. Then, t2 9t 20 (t 4)(t 5) b) 3s3 33s2 54s Begin by factoring out the GCF of 3s. 3s3 33s2 54s 3s(s2 11s 18) 3s(s 2)(s 9) Factor 5t2 18t 8 completely. Sum is 18. 5t2 18t 8 Product: 5 (8) 40 Think of two integers whose product is 40 and whose sum is 18. 20 and 2 www.mhhe.com/messersmith CHAPTER 7 Summary 445
messersmith_power_introductory_algebra_1e_ch4_7_10
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