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This Lesson (The New AC Method to factor trinomials) was created by by tam144(6)
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THE NEW AC METHOD TO FACTOR TRINOMIALS. Factoring a trinomial in standard form f(x) = ax^2 + bx + c means transforming it into two binomials in x. CASE 1. FACTORING TRINOMIAL TYPE x^2 + bx + c (a = 1). The factored trinomial will have the form f(x) = (x + p)(x + q). The new AC Method proceeds to find the 2 numbers p and q that satisfy these 3 conditions: 1) The product (p*q) = a*c (when a= 1, the product is c) 2) The sum (p + q) = b (if the sum = -b, p and q take the opposite values) 3) Application of the Rule of Signs for a quadratic function. Note. The opposite (-p) and (-q) are the 2 real roots of the equation f(x) = 0 RECALL THE RULE OF SIGNS. - If a and c have different signs, p and q have different signs - If a and c have same sign, p and q have same sign. OPERATIONAL METHOD. The New AC Method proceeds by composing factor pairs of (a*c), or c. The pair whose sum equals to (b), or (-b), gives the numbers p and q. Example 1. Factor: f(x) = x^2 + 11x - 102 = (x + p)(x + q) (a*c = c = -102) Solution. The numbers p and q have different signs, since a and c have different signs. Compose factor pairs of c = -102, with all first numbers being negative. Proceed:...(-2, 51)(-3, 34)(-6, 17). This last sum is (-6 + 17) = 11 = b. We get: p = -6 and q = 17. The factoring form is: f(x) = (x - 6)(x + 17). No factoring by grouping. Example 2. Factor: f(x) = x^2 - 28x + 96. (a*c = c = 96). Solution. p and q have same sign. Compose factor pairs of c = 96 with all positive numbers. Proceed:(2, 48)(3, 32)(4, 24). This last sum is (4 + 24) = 28 = -b. Change this sum to the opposite. Then, p = -4 and q = -24. The factoring form is: f(x) = (x - 4)(x - 24). CASE 2. FACTORING TRINOMIALS STANDARD TYPE f(x) = ax^2 + bx + c.(1) The New AC Method proceeds to bring this case to CASE 1. Convert the trinomial f(x) to the trinomial f'(x) that has the form: f'(x) = x^2 + bx + a*c, with a = 1 and constant a*c. Then, we proceed finding the 2 numbers p' and q' exactly like we did in CASE 1. Next, we divide the two numbers p' and q' by the constant a to get p and q by the relations: p = p'/a, and q = q'/a. Example 3. Factor: f(x) = 8x^2 + 22x - 13 (a*c = 8*-13 = -104). Solution. Convert trinomial: f'(x) = x^2 - 22x - 104. Find p' and q' that have different signs. Compose factor pairs of (a*c = -104). Proceed: (-1, 104)(-2, 52)(-4, 26). This last sum is (-4 + 26) = 22 = b. Then, p' = -4 and q' = 26. Next, divide p' and q' by a = 8 to get p and q: p = p'/a = -4/8 = -1/2, and q = q'/a = 26/8 = 13/4. The factoring of original f(x) will be: f(x) = 8(x - 1/2)(x + 13/4). Finally, f(x) = (2x - 1)(4x + 13). Example 4. Factor: f(x) = 12x^2 + 83x + 20. (a*c = 12*20 = 240). Solution. Converted trinomial f'(x) = x^2 + 83x + 240. Find numbers p' and q' that have same sign. Compose fact pairs of a*c = 240. Proceed: (1, 240)(2, 120)(3, 80). This last sum is (3 + 80) = 83 = b. Then p' = 3 and q' = 80. We get p = 3/12 = 1/4, and q = 80/12 = 20/3. The factoring form is f(x) = 12(x + 1/4)(x + 20/3). Finally, f(x) = (4x + 1)(3x + 20). NOTE 1. When composing factor pairs of (a*c), or c, if we can't find the pair whose sum equals to (-b), or (b), then this trinomial can't be factored. NOTE 2. We don't need to write the factor form of the converted trinomial f'(x). We just need to find the 2 numbers p' and q' in order to get the 2 numbers p and q for the original trinomial f(x). PROOF FOR THE NEW AC METHOD. f(x) = ax^2 + bx + c = (1/a)(a^2*x^2 + b*ax + c*a) = (1/a)[(X^2 + bX + a*c)], if we call X = a*x After factoring, like CASE 1, the trinomial (X^2 + bX + a*c)= (X + p')(X + q'), we get: f(x) = (1/a)[(ax + p')(ax + q')] = a(x + p'/a)(x + q'/a) = a(x + p)(x + q) MORE EXAMPLE OF FACTORING TRINOMIAL BY THE NEW AC METHOD. Example 5. Factor: 6x^2 + 17x - 14. (a*c = 6*-14 = -84) Solution. Converted trinomial f'(x) = x^2 + 17x - 84. Find the numbers p' and q' that have different signs. Compose factor pairs of a*c = -84. Proceed: (-1, 84)(-2, 42)(-3, 28)(-4, 21). This last sum is (-4 + 21) = 17 = b. Then, p' = -4, and q' = 21. Back to original f(x), the factoring form is f(x)= 6(x - 2/3)(x + 7/2). Finally, f(x) = (3x - 2)(2x + 7). No factoring by grouping! Example 6. Factor: 6x^2 + 23x + 10. (a*c = 6*10 = 60). Solution. Converted trinomial: f'(x) = x^2 + 23x + 60. Find p' and q' that have the same sign. Compose factor pairs of a*c = 60. Proceed:(2, 30)(3, 20). This last sum is (3 + 20) = 23 = b. Then, p' = 3, and q' = 20. We have p = 3/6 = 1/2, and q = 20/12 = 5/3. The factoring form is f(x) = 6(x + 1/2)(x + 5/3) or f(x) = (2x + 1)(3x + 10). Example 7. Factor: f(x) = 15x^2 - 53x + 16. (a*c = 240) Solution. Converted f'(x) = x^2 - 53x + 240. Numbers p' and q' have same sign. Compose factor pairs of a*c = 240. Proceed: (1, 240)(2, 120)(3, 80) (4, 60) (5, 48). This last sum is 53 = -b. Then, p' = -5 and q' = -48. We deduct: p = -5/15 = -1/3, and q = -48/15 = -16/5. Factoring form: f(x) = 15( x - 1/3)(x - 16/5) or f(x) = (3x - 1)(5x - 16). CONCLUSION. Compared to the existing AC Method, the new AC Method works simpler and faster. Especially, it can avoid the lengthy factoring by grouping. (This article was written by Nghi H Nguyen, the author of the New AC Method for solving quadratic equations - March 2015) This is a sample text of a lesson to show you the capabilities of my lesson system. Try saving this lesson (it will be saved in the development area) and then see how it looks. Then go ahead and change it! Please do not release unfinished lessons to production! Have respect for thousands of children visiting this site and looking for help. Although I am not a control freak, abuse of any kind will not be tolerated. To separate paragraphs, enter a double NEW LINE by pressing the ENTER key twice. 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