Ratio Method for Factoring Trinomials
NAME
DATE
PERIOD
Through the years, several methods have been developed for factoring trinomials whose
leading coefficients are not one. There is trial and error which involves a lot of erasing and
holes rubbed in your paper. There is the grouping method which some people master and
others never quite get. There is the “Tic-Tac-Toe” method which requires you to memorize
what goes in each of the nine boxes on the square. Then there is the ratio method. Here is
how it works.
AS ALWAYS, THE FIRST STEP IN ANY FACTORING PROBLEM IS TO
FACTOR OUT THE GCF.
30x2 + 16x – 24
GCF
2.)
Remember standard form (ax2 + bx + c) and multiply a and c together.
15 ● 12 = 180
3.)
Find factors of the result that will add/subtract to give the middle coefficient b.
Remember: the factors must ADD if the third term is POSITIVE.
Remember: the factors must SUBTRACT if the third term is NEGATIVE.
In this problem we need factors of 180 to SUBTRACT
to give us 8.
FACTORS DIFFERENCE
1
180
179
2
90
88
3
60
57
4
45
41
5
36
31
6
30
24
9
20
11
10
18
8
4.)
Place the leading coefficient a over both of these factors.
15
and 15
10
18
5.)
Reduce the fractions to simplest form.
3
2
6.)
and
=
2(15x2 + 8x – 12)
1.)
5
6
The resulting fractions are what you use to make your binomials. Use these and the
appropriate signs to make the binomials.
Remember: If third term is POSITIVE, you have a DOUBLE SIGN that comes from the
middle term. If third term in NEGATIVE, you have an automatic + / - combo. You will
need to make certain you put the + and – in the correct place.
(3x ▭ 2)(5x ▭ 6)
(3x – 2) (5x + 6)
You can check by FOIL-ing this out!
Thank you to Mrs. Garner who brought this method back from a workshop she attended.
MORE EXAMPLES:
Factor:
1.)
2.)
3.)
4.)
5.)
6.)
Factor:
1.)
2.)
3.)
4.)
5.)
6.)
5x2 + 14x + 8
No GCF
a●c = 5 ● 8 = 40
Factors of 40 that ADD to give 14 are 4 and 10. (Don’t forget, you don’t have to list
every combination unless you declare a trinomial prime.)
5 and 5
4
10
5 and 1
4
2
Third term is positive. Need double sign. Middle term is positive, so double +.
(5x + 4)(x + 2)
4x2 – 15x + 9
No GCF
a●c = 4 ● 9 = 36
Factors of 36 that ADD to give 15 are 3 and 12.
4 and 4
3
12
4 and 1
3
3
Third term is positive. Need double sign. Middle term is negative, so double -.
(4x – 3)(x – 3)
Factor: 12x3 – 26x2 – 10x
1.)
2.)
3.)
4.)
5.)
6.)
GCF = 2x, so 2x(6x2 – 13x – 5)
a●c = 6 ● 5 = 30
Factors of 30 that SUBTRACT to give 13 are 2 and 15.
6 and 6
2
15
3 and 2
1
5
Third term is negative. Automatic + / - combo. Make certain you set it up to get a
NEG 13. Do the OI part of FOIL to check the sign. Do ALL of FOIL to check the
problem. DON’T FORGET THE GCF YOU FACTORED OUT EARLIER!!!
= 2x(3x ▭ 1) (2x ▭ 5)
= 2x(3x + 1) (2x – 5)