AC Factoring of Quadratic
Polynomials
Created by: Patricia Hensley  Jefferson Davis Campus

Factor out the greatest common factor, leaving
ax^{2 }+bx +c.

Find the product of a and c.

Factor this product so that the sum of the
factors is b. That is, find factors d and e of ac such that d
+ e = b.

Write (ax + d) (ax + e).

Factor out and DROP any common factor from
each binomial factor.
EXAMPLE 1: Factor 4x^{2
}– 21x + 5
SOLUTION:

Notice that there is no common factor.

Find the product 4(5) = 20

List factors of 20: (1) (20), (2) (10),
(4) (5)

Find the factors whose sum is –21: 1 and
–20.

Write (4x – 1) (4x – 20).

Drop the common factor from each binomial
factor: (4x –1) (x –5).
EXAMPLE 2: Factor 6x^{2 }–
5x – 6.
SOLUTION:

Notice that there is no common factor.

Find the product 6(6) = 36

A. List factors of 36: (1) (36), (2) (18),
(3) (12), (4) (9), (6) (6)

Find the factors whose sum is –5: 4
and –9.

Write (6x + 4) (6x – 9).

Drop the common factor from each binomial
factor: (3x + 2) (2x – 3).
EXAMPLE 3: Factor 12x^{2 }+
17x – 7.
SOLUTION:

Notice that there is no common factor.

Find the product 12(7) = 84

A. List factors of 84: (1) (84), (4) (21),
(2) (42), (6) (14), (3) (28), (12) (7)

Find the factors whose sum is 17: 4 and 21.

Write (12x – 4) (12x + 21).

Drop the common factor from each binomial
factor: (3x – 1) (4x + 7).
EXAMPLE 4: Factor 8b^{5
}– 68b^{4 }+ 84b^{3}.
SOLUTION:

Take out the common factor: 4b^{3 }(2b^{2}
– 17b + 21).

Find the product: (2) (21) = 42

A. List factors of 42: (1) (42), (2) (21),
(3) (14), (6) (7)

Find the factors whose sum is –17: 3 and
–14.

Write 4b^{3}(2b – 3) (2b – 14).

Drop the common factor from each binomial
factor: 4b^{3} (2b – 3) (b – 7)