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7.5 Factoring Trinomials CORD Math Mrs. Spitz Fall 2006 1 Objectives • Factor quadratic trinomials. 2 of 14 Assignment • Pg. 274 #5-43 all 3 of 14 Introduction • In Lesson 7.1, you learned that when two numbers are multiplied, each number is a factor of the product. Similarly, if two binomials are multiplied, each binomial is a factor of the product. 4 of 14 Consider the binomials 5x + 2 and 3x + 7. You can use the FOIL method to find their products. (5 x 2)(3x 7) (5 x)(3x) (5 x)(7) 2(3x) 2(7) 15 x 35 x 6 x 14 2 15 x 2 (35 6) x 14 15 x 41x 14 2 The binomials 5x + 2 and 3x + 7 are factors of 15x2 + 41x + 14. 5 of 14 FOIL • When using the FOIL method, look at the product of the coefficients of the first and last terms, 15 and 14. Notice that it is the same as the product of the two terms 35 and 6, whose sum is the coefficient of the middle term. You can use this pattern to factor quadratic trinomials, such as 2y2 + 7y +6. 6 of 14 Ex. 1: Factor 2 5x - 17x + 14 • The product of 5 and 14 is 70. Since the product is positive and the sum is negative, both factors must be negative. • Possibilities: Factors of 70 -1, -70 -2, -35 -5, -14 -7, -10 Sum of factors -1+ -70 = -71 NOT -2 + -35 = -37 NOT -5 + -14 = -19 NOT -7 + -10 = -17 YES 7 of 14 Now what? Stop when you find the factors. 5 x 2 [10 (7)] x 14 5 x 2 10 x 7 x 14 (5 x 2 10 x) (7 x 14) 5 x( x 2) (7)( x 2) Factor GCF from each ( x 2)(5 x 7) Factor by grouping Check using FOIL ( x 2)(5 x 7) x 2 7 x 10 x 14 x 2 17 x 14 8 of 14 Ex. 2: What does a factored trinomial look like? What are the factors of 6 - 1 + 6 that subtract to give you 5? Look at the trinomial. Are the signs (x )(x ) positive, negative or both positive and negative. For example: x2 + 5x - 6 6-1=5 One sign is positive, one negative meaning you are looking for factors of 6 that subtract to give you 5—the number in the middle. With no coefficient (the number in front of the variable x2 (the letter), it’s pretty easy to figure out. 9 of 14 Ex. 3: Factor 2 2n -11n + 7 • You must find two numbers whose product is 2 ·7 or 14 and whose sum is -11. Factors of 14 -1, -14 -2, -7 Sum of factors -1+ -14 = -15 -2 + -7 = -9 There are no factors of 14 whose sum is -11. Therefore this expression cannot be factored using integers. It is prime. 10 of 14 Ex. 4: Factor 2 7a + 22a + 3 • You must find two numbers whose product is 7 · 3 or 21 and whose sum is -11. Factors of 21 1, 21 3, 7 Sum of factors 1 + 21 = 22 YES 3 + 7 = 10 NO Factors of 21 that add to give you 22 are 1 and 22. Factored format is (7a+1)(a+3) 11 of 14 Use FOIL to check your answer. a 22a 3 (7 a 1)( a 3) FOIL 2 7 a 21a 1a 3 2 7 a 22a 3 Checks 2 12 of 14 Ex. 5: Factor 2 2q – 9q - 18 • You must find two numbers whose product is 2 · 18 or 36 and whose sum is -9. Factors of 36 -36, 1 -18, 2 -12, 3 Sum of factors -36+1=-35 NOT -18 + 2 = -16 NOT -12 + 3 = -9 YES Factors of -36 that add to give you -9 are -12 and 3. Factored format is (2q + 3)(q – 6) 13 of 14 Use FOIL to check your answer. 2q 9q 18 (2q 3)( q 6) FOIL 2 2q 12q 3q 18 2 2q 9q 18 Checks 2 14 of 14