6.2 Factoring Trinomials
Factoring Trinomials in the Form ax2 + bx + c
The book first considers trinomials in the form x2 + bx + c in which the coefficient of the squared term is 1. In
this case, if the trinomial is factorable, we have two binomial factors.
Rather than learning two methods, I combine both of first two sections and let the coefficient of the square
term to be any real number except 0.
Note sometimes we can also factor a monomial only from all terms.
Remember FOIL
Simple example:
(x + 2)(x + 3) = x2 + 3x + 2x + 6
= x2 + 5x + 6
Note: The coefficient of the middle term is ad + bc and the last term is bd.
Factoring by the “ac” Method
To factor a trinomial of the form ax2 + bx + c, (where a ≠ 0) using the ac method:
1. Look for a monomial GCF in all of the terms. If there is one, factor it out first.
2. Find two factors of the product ac whose sum is b.
3. Write a four-term polynomial in which bx is written as the sum of two like terms whose coefficients are the two
factors you found in step 2.
4. Factor by grouping.
Positive or Negative
If a < 0, factor out -1, so a > 0 (assume both b ≠ 0 and c ≠ 0)
If ac > 0
if b > 0, both factors are positive
if b < 0, both factors are negative
If ac < 0
one factor is postive, the other is negative
Examples
[08]
x2 + 12x + 11
no common factor
ac = 11, b = 12
factors
ac b
-11, -1
11 -12
11, 1
11 12 ok
x2 + 11x + 1x + 11
x(x + 11) + 1(x + 11)
(x + 11) (x + 1)
[18]
x2 - x - 30
no common factor
ac = -30, b = -1
factors
ac b
-10, 3
-30 -7
5,-6
-30 -1 ok
x2 + 5x – 6x - 30
x(x + 5) - 6(x + 5)
(x + 5) (x - 6)
[24]
7x3 – 35x2 + 28x
GCF = 7x
7x( x2 – 5x + 4 )
ac = 4 b = -5
factors
ac b
-4, -1
4 -5 ok
7x( x2 – 4x – x + 4 )
7x{ x(x – 4) – 1(x – 4)}
7x (x – 4) (x – 1)
[30]
x2 – 3x - 12
no common factor
ac = -12, b = -3
factors
ac b
-12, 1
-12 -11
-6, 2
-12 -4
-4, 3
-12 -1
nothing will work DOES NOT FACTOR
[40]
[50]
[68]
[82]
[90]
2x2 + 5x - 3
no common factor
ac = -6, b = 5
factors
ac b
-3, 2
-6 -1
-6, 1
-6 -5
6, -1
-6 5 ok
2x2 + 6x – 1x - 3
2x(x + 3) - 1(x + 3)
(x + 3) (2x - 1)
6x2 - 7x – 10
no common factor
ac = -60, b = -7
factors
ac b
-3, 20
-60 17
-6, 10
-60 4
-5, 12
-60 7
5, -12
-60 -7 ok
6x2 + 5x – 12x - 10
x(6x + 5) - 2(6x + 5)
(6x + 5) (x - 2)
24x4 - 80x3 – 64x2
GCF = 8x2
2
8x (3x2 – 10x – 8)
ac = -24, b = -10
factors
ac b
-3, 8
-24 5
-2, 12
-24 10
-12, 2
-24 -10 ok
8x2 (3x2 - 12x + 2x – 8)
8x2 {3x(x - 4) + 2(x - 4)}
8x2 (x - 4) (3x + 2)
8x4 + 2x2 – 3
no common factors
let a = x2
8a2 + 2a - 3
ac = -24, b = 2
factors
ac b
-3, 8
-24 5
-2, 12
-24 10
-4, 6
-24 2 ok
8a2 – 4a + 6a – 3
4a(2a – 1) + 3(2a – 1)
(2a – 1) (4a + 3)
(2x2 – 1)(4x2 + 3)
2(x-3)2 + 7(x-3) + 6
no common factors
let a = x-3
2a2 + 7a + 6
ac = 12, b = 7
factors
ac b
6, 2
12 8
4, 3
12 7 ok
2a2 + 4a + 3a + 6
2a(a + 2) + 3(a + 2)
(a + 2) (2a + 3)
({x-3} + 2)(2{x-3} + 3)
( x – 1 )( 2x – 3 )