How do you factor by grouping #28s^2 - 37s - 21#?

1 Answer
Apr 23, 2015

Factoring #f(x) = 28x^2 - 37x - 21#.

There are 2 methods.

1. Factoring AC method (factoring by grouping) .
Find 2 numbers b1 and b2 that satisfy these 2 conditions:
Sum: #(b1 + b2) = -37#
and
Product: # (b1*b2) = a*c = 588#

To find #b1 " and "b2" "# compose factor pairs of #a*c = -588#
Proceed: #(1, -588),(2,-294),....(12, -49)..."etc."#
We find
#b1 = 12" and "b2 = -49#
( since their sum is #-37#. )

Next factor by grouping:
#f(x) = 28x^2 - 49x + 12x - 21#
# = 7x*(4x - 7) + 3*(4x - 7)#

Factored form:
f(x) = (4x - 7)(7x + 3)

2. The new AC method to factor a trinomial f(x)
#f(x) = 28x^2 - 37x - 21." (1)"#

First convert trinomial (1) to
trinomial: #f(x) = x^2 - 37x - 588 " (2)#,
with #a*c = -21*(28) = -588#

Compose factor pairs of #a*c = -#
and apply the Rule of Sign for Real Roots.

Proceed: #(-1, 588)....(-12, 49)#

This sum is #49 - 12 = 37 = -b#

Then #b'1 = 12" and #b'2 = -49#

Next, divide #b'1" and "b'2# by a
to get #b1" and "b2# for trinomial (1).

#b1 = b'1/a = 12/28 = 3/7 ,#
and
#b2 = b'2/a = -49/28 = -7/4.#

Then, the factored form is:
#f(x) = (x + 3/7)(x - 7/4)#
# = (7x + 3)(4x - 7)#

This new AC Method avoids the lengthy factoring by grouping.