Factoring Trinomials: The ac Method


 Pamela Preston
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1 6.7 Factoring Trinomials: The ac Method 6.7 OBJECTIVES 1. Use the ac test to determine whether a trinomial is factorable over the integers 2. Use the results of the ac test to factor a trinomial 3. For a given value of x, evaluate f(x) before and after factoring The product of two binomials of the form ( x )( x ) will be a trinomial. In your earlier mathematics classes, you used the FOIL method to find the product of two binomials. In this section, we will use the factoring by grouping method to find the binomial factors for a trinomial. First, let s look at some factored trinomials. Example 1 Matching Trinomials and Their Factors Determine which of the following are true statements. (a) x 2 2x 8 (x 4)(x 2) This is a true statement. Using the FOIL method, we see that (x 4)(x 2) x 2 2x 4x 8 x 2 2x 8 x 2 6x 5 (x 2)(x 3) Not a true statement, because (x 2)(x 3) x 2 3x 2x 6 x 2 5x 6 (c) x 2 5x 14 (x 2)(x 7) True, because (x 2)(x 7) x 2 7x 2x 14 x 2 5x 14 (d) x 2 8x 15 (x 5)(x 3) False, because (x 5)(x 3) x 2 3x 5x 15 x 2 8x 15 CHECK YOURSELF 1 Determine which of the following are true statements. (a) 2x 2 2x 3 (2x 3)(x 1) 3x 2 11x 4 (3x 1)(x 4) (c) 2x 2 7x 3 (x 3)(2x 1) 431
2 432 CHAPTER 6 POLYNOMIALS AND POLYNOMIAL FUNCTIONS The first step in learning to factor a trinomial is to identify its coefficients. To be consistent, we first write the trinomial in standard ax 2 bx c form, then label the three coefficients as a, b, and c. Example 2 Identifying the Coefficients of ax 2 bx c When necessary, rewrite the trinomial in ax 2 bx c form. Then label a, b, and c. (a) x 2 3x 18 a 1 b 3 c 18 x 2 24x 23 a 1 b 24 c 23 (c) x x First rewrite the trinomial in descending order. x 2 11x 8 Then, a 1 b 11 c 8 CHECK YOURSELF 2 When necessary, rewrite the trinomial in ax 2 bx c form. Then label a, b, and c. (a) x 2 5x 14 x 2 18x 17 (c) x 6 2x 2 Not all trinomials can be factored.to discover if a trinomial is factorable, we try the ac test. Rules and Properties: The ac Test A trinomial of the form ax 2 bx c is factorable if (and only if) there are two integers, m and n, such that ac mn and b m n In Example 3, we will determine whether each trinomial is factorable by finding the values of m and n. Example 3 Using the ac Test Use the ac test to determine which of the following trinomials can be factored. Find the values of m and n for each trinomial that can be factored. (a) x 2 3x 18 First, we note that a 1, b 3, and c 18, so ac 1( 18) 18. Then, we look for two numbers, m and n, such that mn ac and m n b. In this case, that means mn 18 m n 3
3 FACTORING TRINOMIALS: THE ac METHOD SECTION We will look at every pair of integers with a product of 18. We then look at the sum of each pair. mn 1( 18) 18 2( 9) 18 3( 6) 18 m n 1 ( 18) 17 2 ( 9) 7 3 ( 6) 3 We need look no further because we have found two integers whose mn product is 18 and m n sum is 3. m 3 n 6 x 2 24x 23 We see that a 1, b 24, and c 23. So, ac 23 and b 24. Therefore, we want mn 23 m n 24 We now work with integer pairs, looking for two integers with a product of 23 and a sum of 24. mn m n 1(23) ( 23) 23 1 ( 23) 24 We find that m 1 and n 23. (c) x 2 11x 8 We see that a 1, b 11, and c 8. So, ac 8 and b 11. Therefore, we want mn 8 m n 11 mn m n 1(8) (4) ( 8) 8 1 ( 8) 9 2( 4) 8 2 ( 4) 6 There is no other pair of integers with a product of 8, and none has a sum of 11. The trinomial x 2 11x 8 is not factorable. (d) 2x 2 7x 15 We see that a 2, b 7, and c 15. So, ac 30 and b 7. Therefore, we want mn 30 m n 7 mn m n 1( 30) 30 1 ( 30) 29 2( 15) 30 2 ( 15) 13 3( 10) 30 3 ( 10) 7 5( 6) 30 5 ( 6) 1 6( 5) 30 6 ( 5) 1 10( 3) ( 3) 7
4 434 CHAPTER 6 POLYNOMIALS AND POLYNOMIAL FUNCTIONS There is no need to go further. We have found two integers with a product of 30 and a sum of 7. So m 10 and n 3. In this example, you may have noticed patterns and shortcuts that make it easier to find m and n. By all means, use those patterns. This is essential in mathematical thinking. You are taught a stepbystep process that will always work for solving a problem; this process is called an algorithm. It is very easy to teach a computer an algorithm. It is very difficult (some would say impossible) for a computer to have insight. Shortcuts that you discover are insights. They may be the most important part of your mathematical education. CHECK YOURSELF 3 Use the ac test to determine which of the following trinomials can be factored. Find the values of m and n for each trinomial that can be factored. (a) x 2 7x 12 x 2 5x 14 (c) 2x 2 x 6 (d) 3x 2 6x 7 So far we have used the results of the ac test only to determine whether a trinomial is factorable. The results can also be used to help factor the trinomial. Example 4 Using the Results of the ac Test to Factor a Trinomial Rewrite the middle term as the sum of two terms, then factor by grouping. (a) x 2 3x 18 We see that a 1, b 3, and c 18, so ac 18 b 3 We are looking for two numbers, m and n, so that mn 18 m n 3 In Example 3, we found that the two integers were 3 and 6 because 3( 6) 18 and 3 ( 6) 3. That result is used to rewrite the middle term (here 3x) as the sum of two terms. We now rewrite the middle term as the sum of 3x and 6x. x 2 3x 6x 18 Then, we factor by grouping: x 2 3x 6x 18 x(x 3) 6(x 3) (x 3)(x 6) x 2 24x 23 We use the results of Example 3, in which we found m 1 and n 23, to rewrite the middle term of the expression. x 2 24x 23 x 2 x 23x 23 Then, we factor by grouping: x 2 x 23x 23 x(x 1) 23(x 1) (x 1)(x 23)
5 FACTORING TRINOMIALS: THE ac METHOD SECTION (c) 2x 2 7x 15 From example 3(d ), we know that this trinomial is factorable and that m 10 and n 3. We use that result to rewrite the middle term of the trinomial. 2x 2 7x 15 2x 2 10x 3x 15 2x(x 5) 3(x 5) (x 5)(2x 3) CHECK YOURSELF 4 Rewrite the middle term as the sum of two terms, then factor by grouping. (a) x 2 7x 12 x 2 5x 14 (c) 2x 2 x 6 (d) 3x 2 7x 6 Not all product pairs need to be tried to find m and n. A look at the sign pattern will eliminate many of the possibilities. Assuming the lead coefficient to be positive, there are four possible sign patterns. Pattern Example Conclusion 1. b and c are both positive. 2x 2 13x 15 m and n must be positive. 2. b is negative and c is x 2 3x 2 m and n must both be negative. positive. 3. b is positive and c is x 2 5x 14 m and n are of opposite signs. negative. (The value with the larger absolute value is positive.) 4. b and c are both negative. x 2 4x 4 m and n are of opposite signs. (The value with the larger absolute value is negative.) Sometimes the factors of a trinomial seem obvious. At other times you might be certain that there are only a couple of possible sets of factors for a trinomial. It is perfectly acceptable to check these proposed factors to see if they work. If you find the factors in this manner, we say that you have used the trial and error method. This method is discussed in Section 6.7*, which follows this section. To this point we have been factoring polynomial expressions. When a function is defined by a polynomial expression, we can factor that expression without affecting any of the ordered pairs associated with the function. Factoring the expression makes it easier to find some of the ordered pairs. In particular, we will be looking for values of x that cause f(x) to be 0. We do this by using the zero product rule. Rules and Properties: Zero Product Rule If 0 ab, then either a 0, b 0, or both are zero. Another way to say this is, if the product of two numbers is zero, then at least one of those numbers must be zero.
6 436 CHAPTER 6 POLYNOMIALS AND POLYNOMIAL FUNCTIONS Example 5 Factoring Polynomial Functions Given the function f(x) 2x 2 7x 15, complete the following. (a) Rewrite the function in factored form. From Example 4(c) we have f(x) (x 5)(2x 3) Find the ordered pair associated with f(0). f(0) (0 5)(0 3) 15 The ordered pair is (0, 15). (c) Find all ordered pairs (x, 0). We are looking for the x value for which f(x) 0, so 0 (x 5)(2x 3) By the zero product rule, we know that either (x 5) 0 or (2x 3) 0 which means that x 5 or x 3 2 The ordered pairs are ( 5, 0) and 3 2, 0. ordered pairs are associated with that function. Check the original function to see that these CHECK YOURSELF 5 Given the function f(x) 2x 2 x 6, complete the following. (a) Rewrite the function in factored form. Find the ordered pair associated with f(0). (c) Find all ordered pairs (x, 0). CHECK YOURSELF ANSWERS 1. (a) False; true; (c) true 2. (a) a 1, b 5, c 14; a 1, b 18, c 17; (c) a 2, b 1, c 6 3. (a) Factorable, m 4, n 3; factorable, m 7, n 2; (c) factorable, m 4, n 3; (d) not factorable 4. (a) x 2 3x 4x 12 (x 3)(x 4); x 2 7x 2x 14 (x 7)(x 2); (c) 2x 2 x 6 2x 2 4x 3x 6 (2x 3)(x 2); (d) 3x 2 7x 6 3x 2 9x 2x 6 (3x 2)(x 3) 3 2, 0 5. (a) f(x) (2x 3)(x 2); (0, 6); (c) and (2, 0)
7 Name 6.7 Exercises Section Date In exercises 1 to 8, determine which are true statements. 1. x 2 2x 3 (x 1)(x 3) ANSWERS x 2 2x 8 (x 2)(x 4) 3. 2x 2 5x 4 (2x 1)(x 4) 4. 3x 2 13x 10 (3x 2)(x 5) 5. x 2 x 6 (x 5)(x 1) 6. 6x 2 7x 3 (3x 1)(2x 3) 7. 2x 2 11x 5 ( x 5)(2x 1) 8. 6x 2 13x 6 (2x 3)( 3x 2) In exercises 9 to 16, when necessary, rewrite the trinomial in ax 2 bx c form, then label a, b, and c. 9. x 2 3x x 2 2x x 2 5x x 2 x x 1 2x x 3x x 3x x x In exercises 17 to 24, use the ac test to determine which trinomials can be factored. Find the values of m and n for each trinomial that can be factored. 17. x 2 3x x 2 x x 2 2x x 2 7x
8 ANSWERS x 2 3x x 2 10x x 2 5x x 2 x In exercises 25 to 70, completely factor each polynomial expression. 25. x 2 7x x 2 9x x 2 9x x 2 11x x 2 15x x 2 13x x 2 7x x 2 7x x 2 10x x 2 13x x 2 7x x 2 15x x 2 8xy 15y x 2 9xy 20y x 2 16xy 55y x 2 9xy 22y x 2 11x x 2 9x x 2 18x x 2 20x x 2 23x x 2 30x 7 438
9 ANSWERS 47. 4x 2 20x x 2 24x x 2 19x x 2 17x x 2 24x x 2 14x x 2 7x x 2 5x x 2 40x x 2 45x x 2 17xy 6y x 2 17xy 12y x 2 30xy 7y x 2 14xy 15y x 2 24x x 2 10x x 2 26x x 2 39x x 3 31x 2 5x 66. 8x 3 25x 2 3x 67. 5x 3 14x 2 24x 68. 3x 4 17x 3 28x x 3 15x 2 y 18xy x 3 10x 2 y 72xy
10 ANSWERS 71. (a) (c) 72. (a) (c) 73. (a) (c) In exercises 71 to 76, for each function, (a) rewrite the function in factored form, find the ordered pair associated with f(0), and (c) find all ordered pairs (x, 0). 71. f(x) x 2 2x f(x) x 2 3x f(x) 2x 2 3x f(x) 3x 2 11x (a) (c) (a) (c) (a) (c) 75. f(x) 3x 2 5x f(x) 10x 2 13x 3 Certain trinomials in quadratic form can be factored with similar techniques. For instance, we can factor x 4 5x 2 6 as (x 2 6)(x 2 1). In exercises 77 to 88, apply a similar method to completely factor each polynomial. 77. x 4 3x x 4 7x x 4 8x x 4 5x y 6 2y x 6 10x x 5 6x 3 16x 84. x 6 8x 4 15x x 4 5x x 4 5x x 6 6x x 6 2x 3 3 In exercises 89 to 96, determine a value of the number k so that the polynomial can be factored. 89. x 2 5x k 90. x 2 3x k 440
11 ANSWERS 91. 6x 2 x k 92. 4x 2 x k 93. x 2 kx x 2 kx x 2 kx x 2 kx The product of three numbers is x 3 6x 2 8x. Show that the numbers are consecutive even integers. (Hint: Factor the expression.) 98. The product of three numbers is x 3 3x 2 2x. Show that the numbers are consecutive integers (a) 100. (a) 101. (a) In each of the following, (a) factor the given function, identify the values of x for which f(x) 0, (c) graph f(x) using the graphing calculator and determine where the graph crosses the x axis, and (d) compare the results of and (c). 99. f(x) x 2 2x f(x) x 2 3x (a) 101. f(x) 2x 2 x f(x) 3x 2 x 2 In exercises 103 and 104, determine the binomials that represent the dimensions of the given figure Area 2x 2 7x 15? Area 3x 2 11x 10??? 441
12 Answers 1. True 3. False 5. False 7. False 9. a 1, b 3, c a 2, b 5, c a 2, b 1, c a 3, b 2, c Factorable, m 5, n Not factorable 21. Not factorable 23. Factorable, m 4, n (x 3)(x 4) 27. (x 8)(x 1) 29. (x 10)(x 5) 31. (x 10)(x 3) 33. (x 6)(x 4) 35. (x 11)(x 4) 37. (x 3y)(x 5y) 39. (x 11y)(x 5y) 41. (3x 4)(x 5) 43. (5x 2)(x 4) 45. (3x 5)(4x 1) 47. (2x 5) (5x 6)(x 5) 51. (5x 6)(x 6) 53. (2x 3)(5x 4) 55. (4x 5) (7x 3y)(x 2y) 59. (4x y)(2x 7y) 61. 3(x 5)(x 3) 63. 2(x 4)(x 9) 65. x(6x 1)(x 5) 67. x(x 4)(5x 6) 69. 3x(x 6y)(x y) 71. (a) (x 3)(x 1); (0, 3); (c) (3, 0) and ( 1, 0) 1 2, (a) (2x 1)(x 2); (0, 2); (c) and ( 2, 0) 75. (a) (x 4)(3x 7); (0, 28); (c) ( 4, 0) and 7 3, (x 2 1)(x 2 2) 79. (x 2 11)(x 2 3) 81. (y 3 5)(y 3 3) 83. x(x 2 2)(x 2 8) 85. (x 3)(x 3)(x 2 4) 87. (x 2)(x 2 2x 4)(x 3 2) , 1, or , 5, 1, or , 7, 17, 17, 3, or x(x 2)(x 4) (a) (x 4)(x 2); 4, (a) (2x 3)(x 1); 1 2, 103. (2x 3) and (x 5) 442
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