Case 1. Factoring trinomial type f(x) = x^2 + bx + cf(x)=x2+bx+c.
The factored trinomial will have the form: f(x) = (x + p)(x + q)f(x)=(x+p)(x+q).
The new AC Method finds 22 numbers p and qpandq that satisfy these 3 conditions:
- The product p*q = a*cp⋅q=a⋅c. (When a = 1a=1, this product is cc)
- The sum (p + q) = b(p+q)=b
- Application of the rule of Signs for real roots.
Reminder of the Rule of Signs.
- When a and caandc have different signs, p and qpandq have opposite signs.
- When a and caandc have the same sign, p and qpandq have the same sign.
New AC Method.
To find p and qpandq, compose factor pairs of cc, and in the same time, apply the Rule of Signs . The pair whose sum equals to (-b)(−b), or (b)(b), gives p and qpandq.
Example 1. Factor f(x) = x^2 + 31x + 108. f(x)=x2+31x+108.
Solution. p and qpandq have the same sign. Compose factor pairs of c = 108c=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.
