Question

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

Answer 1

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.