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. 358) Factor out the greatest common factor. 6 40d 16d 5 72d 4 The GCF is 8d 4. 16d 6 40d 5 72d 4 (8d 4)(2d 2) (8d 4)(5d) (8d 4)(9) 4(2d 8d 2 5d 9) 4(2d Check: 8d 2 5d 9) 16d 6 40d 5 72d 4 ✓ 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. 363) 7.2 Factoring Trinomials 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. 367) Factoring ax2 bx c by Grouping (p. 370) Factor completely. 45tu 27t 20u 12 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. 45tu 27t 20u 12 T T 9t(5u 3) 4(5u 3) Take out the common factor. (5u 3)(9t 4) Factor out (5u 3). Check: 15u 32 19t 42 45tu 27t 20u 12 ✓ Factor completely. a) y2 7y 12 Think of two numbers whose product is 12 and whose sum is 7. 3 and 4. Then, y2 7y 12 (y 3)(y 4) b) 2r3 26r2 60r Begin by factoring out the GCF of 2r. 2r3 26r2 60r 2r(r2 13r 30) 2r(r 10)(r 3) Factor completely. 5n2 18n 8 Sum is 18 T 5n2 18n 8 Product: 5 (8) 40 Think of two integers whose product is 40 and whose sum is 18. 20 and 2 414 CHAPTER 7 Factoring Polynomials www.mhhe.com/messersmith
messersmith_power_intermediate_algebra_1e_ch4_7_10
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