What is the new AC Method to factor trinomials?

1 Answer

Answer:

Use the new AC Method.

Explanation:

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:

  1. The product #p*q = a*c#. (When #a = 1#, this product is #c#)
  2. The sum #(p + q) = b#
  3. Application of the rule of Signs for real roots.

Reminder of the Rule of Signs.

  • When #a and c# have different signs, #p and q# have opposite signs.
  • When #a and c# have the same sign, #p and q# have the same sign.

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 factoring by grouping.