22 Jun Use the new AC Method. Case 1. Factoring trinomial type ##f(
Use the new AC Method. Case 1. Factoring trinomial type ##f(x) = x^2 + bx + c##. The factored trinomial will have the form: ##f(x) = (x + p)(x + q)##. The new AC Method finds ##2## numbers ##p and q## that satisfy these 3 conditions: Reminder of the Rule of Signs. New AC Method. To find ##p and q## compose factor pairs of ##c## and in the same time apply the Rule of Signs . The pair whose sum equals to ##(-b)## or ##(b)## gives ##p and q##. Example 1. Factor ##f(x) = x^2 + 31x + 108. ## Solution. ##p and q## have the same sign. Compose factor pairs of ##c = 108##. Proceed: ##…(2 54) (3 36) (4 27)##. The last sum is ##4 + 27 = 31 = b##. Then ##p = 4 and q = 27##. Factoring form: ##f(x) = (x + 4)(x + 27)## .CASE 2 . Factor trinomial standard type ##f(x) = ax^2 + bx + c## (1) Bring back to Case 1. Convert ##f(x)## to ##f'(x) = x^2 + bx + a*c = (x + p’)(x + q’)##. Find ##p’ and q’## by the method mentioned in Case 1 . Then divide ##p’ and q’## by ##(a)## to get ##p and q## for trinomial (1). Example 2 . Factor ##f(x) = 8x^2 + 22x – 13 = 8(x + p)(x + q)## (1). Converted trinomial: ##f'(x) = x^2 + 22x – 104 = (x + p’)(x + q’)## (2). ##p’ and q’## have opposite signs. Compose factor pairs of ##(ac = -104) –> … (-2 52) (-4 26)##. This last sum is ##(26 – 4 = 22 = b)##. Then ##p’ = -4 and q’ = 26##. Back to the original trinomial (1): ##p = (p’)/a = -4/8 = -1/2 and q = (q’)/a = 26/8 = 13/4##. Factoring form ##f(x) = 8(x – 1/2)(x + 13/4) = (2x – 1)(4x + 13).## This new AC Method avoids the lengthy .
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