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CHAPTER 5 Polynomials: Factoring Slide 2Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. 5.1Introduction to Factoring 5.2Factoring Trinomials of the Type x 2 + bx + c 5.3Factoring ax 2 + bx + c, a ≠ 1: The FOIL Method 5.4Factoring ax 2 + bx + c, a ≠ 1: The ac-Method 5.5Factoring Trinomial Squares and Differences of Squares 5.6Factoring: A General Strategy 5.7Solving Quadratic Equations by Factoring 5.8Applications of Quadratic Equations

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OBJECTIVES 5.3 Factoring ax 2 + bx + c, a ≠ 1: The FOIL Method Slide 3Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. aFactoring ax 2 + bx + c, a ≠ 1, using the FOIL method.

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5.3 Factoring ax 2 + bx + c, a ≠ 1: The FOIL Method The FOIL Method (continued) Slide 4Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

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5.3 Factoring ax 2 + bx + c, a ≠ 1: The FOIL Method The FOIL Method Slide 5Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

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EXAMPLE Solution 1. First, check for a common factor. There is none other than 1 or 1. 2. Find the First terms whose product is 3x 2. The only possibilities are 3x and x: (3x + )(x + ) 5.3 Factoring ax 2 + bx + c, a ≠ 1: The FOIL Method a Factoring ax 2 + bx + c, a ≠ 1, using the FOIL method. A Factor: 3x 2 14x 5 (continued) Slide 6Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

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EXAMPLE Solution 3. Find the Last terms whose product is 5. Possibilities are ( 5)(1), (5)( 1) Important!: Since the First terms are not identical, we must also consider the above factors in reverse order: (1)( 5), and ( 1)(5). 5.3 Factoring ax 2 + bx + c, a ≠ 1: The FOIL Method a Factoring ax 2 + bx + c, a ≠ 1, using the FOIL method. A Factor: 3x 2 14x 5 (continued) Slide 7Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

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EXAMPLE 4. Knowing that the First and Last products will check, inspect the Outer and Inner products resulting from steps (2) and (3) Look for the combination in which the sum of the products is the middle term 5.3 Factoring ax 2 + bx + c, a ≠ 1: The FOIL Method a Factoring ax 2 + bx + c, a ≠ 1, using the FOIL method. A Factor: 3x 2 14x 5 (continued) Slide 8Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

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EXAMPLE (3x 5)(x + 1) = 3x 2 + 3x 5x 5 = 3x 2 2x 5 (3x 1)(x + 5) = 3x 2 + 15x x 5 = 3x 2 + 14x 5 (3x + 5)(x 1) = 3x 2 3x + 5x 5 = 3x 2 + 2x 5 (3x + 1)(x 5) = 3x 2 15x + x 5 = 3x 2 14x 5 Wrong middle term Correct middle term! 5.3 Factoring ax 2 + bx + c, a ≠ 1: The FOIL Method a Factoring ax 2 + bx + c, a ≠ 1, using the FOIL method. A Factor: 3x 2 14x 5 Slide 9Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

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EXAMPLE Solution 1. First, factor out the largest common factor, 3x: 3x(6x 2 47x 8) 2. Factor 6x 2 47x 8. Since 6x 2 can be factored as 3x 2x or 6x x, we have two possibilities (3x + )(2x + ) or (6x + )(x + ) 5.3 Factoring ax 2 + bx + c, a ≠ 1: The FOIL Method a Factoring ax 2 + bx + c, a ≠ 1, using the FOIL method. B Factor: 18x 3 141x 2 24x (continued) Slide 10Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

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EXAMPLE Solution 3. There are several pairs of factors of 8. List each way: 8, 11, 8 2, 4 4, 2 8, 1 1, 8 2, 4 4, 2 5.3 Factoring ax 2 + bx + c, a ≠ 1: The FOIL Method a Factoring ax 2 + bx + c, a ≠ 1, using the FOIL method. B Factor: 18x 3 141x 2 24x (continued) Slide 11Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

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EXAMPLE TrialProduct (6x + 4)(x 2)6x 2 12x + 4x 8 = 6x 2 8x 8 (6x 4)(x + 2)6x 2 + 12x 4x 8 = 6x 2 + 8x + 8 (6x + 1)(x 8)6x 2 48x + x 8 = 6x 2 47x 8 We do not need to consider (3x + )(2x + ). The complete factorization is 3x(6x + 1)(x 8). Wrong middle term Correct middle term 5.3 Factoring ax 2 + bx + c, a ≠ 1: The FOIL Method a Factoring ax 2 + bx + c, a ≠ 1, using the FOIL method. B Factor: 18x 3 141x 2 24x Slide 12Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

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STUDY TIP Tips for Factoring ax 2 + bx + c Always factor out the largest common factor, if one exists. Once the largest common factor has been factored out of the original trinomial, no binomial factor can contain a common factor (other than 1 or –1). If c is positive, then the signs in both binomial factors must match the sign of b. 5.3 Factoring ax 2 + bx + c, a ≠ 1: The FOIL Method Slide 13Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

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STUDY TIP Reversing the signs in the binomials reverses the sign of the middle term of their product. Organize your work so that you can keep track of which possibilities you have checked. Always check by multiplying. 5.3 Factoring ax 2 + bx + c, a ≠ 1: The FOIL Method Slide 14Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

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EXAMPLE Solution: An important problem-solving strategy is to find a way to make problems look like problems we already know how to solve. Rewrite the equation in descending order. 14x + 5 3x 2 = 3x 2 + 14x + 5 Factor out the 1: 3x 2 + 14x + 5 = 1(3x 2 14x 5) = 1(3x + 1)(x 5) The factorization of 14x + 5 3x 2 is 1(3x + 1)(x 5). 5.3 Factoring ax 2 + bx + c, a ≠ 1: The FOIL Method a Factoring ax 2 + bx + c, a ≠ 1, using the FOIL method. C Factor: 14x + 5 3x 2 Slide 15Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

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EXAMPLE Solution: No common factors exist, we examine the first term, 6x 2. There are two possibilities: (2x + )(3x + ) or (6x + )(x + ). The last term 12y 2, has the following pairs of factors: 12y, y6y, 2y4y, 3y and 12y, y 6y, 2y 4y, 3y as well as each of the pairings reversed. 5.3 Factoring ax 2 + bx + c, a ≠ 1: The FOIL Method a Factoring ax 2 + bx + c, a ≠ 1, using the FOIL method. D Factor: 6x 2 xy 12y 2 (continued) Slide 16Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

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EXAMPLE Some trials like (2x 6y)(3x + 2y) and (6x + 4y) (x 3y), cannot be correct because (2x 6y) and (6x + 4y) contain a common factor, 2. 5.3 Factoring ax 2 + bx + c, a ≠ 1: The FOIL Method a Factoring ax 2 + bx + c, a ≠ 1, using the FOIL method. D Factor: 6x 2 xy 12y 2 (continued) Slide 17Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

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EXAMPLE TrialProduct (2x + 3y)(3x 4y)6x 2 8xy + 9xy 12y 2 = 6x 2 + xy 12y 2 Our trial is incorrect, but only because of the sign of the middle term. To correctly factor, simply change the signs in the binomials. (2x 3y)(3x + 4y)6x 2 + 8xy 9xy 12y 2 = 6x 2 xy 12y 2 The correct factorization is (2x 3y)(3x + 4y). 5.3 Factoring ax 2 + bx + c, a ≠ 1: The FOIL Method a Factoring ax 2 + bx + c, a ≠ 1, using the FOIL method. D Factor: 6x 2 xy 12y 2 Slide 18Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

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