SYNTHETIC FACTORING:
A STUDENT CENTERED APPROACH
IN FACTORING TRINOMIALS
Jon Anderson
Darren Wiberg

FACTORING
• Factoring is a difficult concept for Beginning and Intermediate Algebra students to understand and master
• In particular, students can find it difficult to factor trinomials that do not begin with a 1.
• Many methods already exist that attempt to teach students how to factor these trinomials
• Trial and Error
• Factoring by Grouping
• The Box Method
• And others
• While these methods work, we have found that students struggle to learn and master using them

INTRODUCING SYNTHETIC FACTORING
• Synthetic Factoring has the following advantages for students:
• Takes advantage of a student’s prior knowledge of simplifying fractions
• Can be done quickly while still showing all work
• Can accurately find factors of difficult trinomials
• Can find the zeros of factorable quadratic polynomials quickly
• Can be used to factor any factorable trinomial, allowing students to learn only a single, easy to learn method
• Provides a mathematically sound method for factoring they can use in future math courses
• Increases students’ confidence in their ability to factor

INTRODUCING SYNTHETIC FACTORING
• Synthetic Factoring also benefits the teacher in many ways. It allows teachers:
• the ability to show all work while factoring any trinomial, including those found in complicated rational expressions
• to work through more examples in class with students following along
• to teach a single method for factoring
• to cover factoring trinomials in a single lecture period without harming the students’ ability to learn
• to teach a mathematically sound method

INTRODUCING SYNTHETIC FACTORING
• So, what is Synthetic Factoring? Here are the steps to use:
• Given a trinomial in the form of 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 with 𝑎 , 𝑏 , and 𝑐 having no common factors. If 𝑎 , 𝑏 , and 𝑐 have a common factor, factor it out, then begin with step 1 on the resulting trinomial.
1.
Find the value of 𝑎𝑐 .
2.
Find a pair of integers 𝑚 and 𝑛 where 𝑚𝑛 = 𝑎𝑐 and 𝑚 + 𝑛 = 𝑏 .
3.
Create fractions of the form 𝑚 𝑎 and 𝑛 𝑎
.
4.
Simplify the fractions formed in step 3 as 𝑚 𝑎
= 𝑞 𝑝 and 𝑛 𝑎
= 𝑠 𝑟
.
5.
Form the factors 𝑝𝑥 + 𝑞 𝑟𝑥 + 𝑠 .
6.
Check the answer by FOILing the result

STUDENT THOUGHTS ON
SYNTHETIC FACTORING
• Students have responded positively to Synthetic Factoring
• In an initial survey performed in Fall 2010 of 54 students found in
Beginning and Intermediate Algebra:
• 36 said they favored Synthetic Factoring
• 6 said they favored Trial and Error
• 9 said they favored Factoring by Grouping

STUDENT THOUGHTS ON
SYNTHETIC FACTORING
• Of the 36 who favored Synthetic Factoring:
• 25 said they had a high degree of confidence in their ability to factor
• 6 said they had a decent degree of confidence in their ability to factor
• 1 said they had a low degree of confidence in their ability to factor
• Of the 6 students who favored Trial and Error
• 5 said they had a decent degree of confidence in their ability to factor
• 1 said they had a low degree of confidence in their ability to factor
• Of the 9 students who favored Factoring by Grouping
• 3 said they had a high degree of confidence in their ability to factor
• 4 said they had a decent degree of confidence in their ability to factor
• 2 said they had a low degree of confidence in their ability to factor

STUDENT THOUGHTS ON
SYNTHETIC FACTORING
• Quotations from students after being shown Synthetic Factoring in Fall
2010:
 “The factoring method I use is the [Synthetic Factoring] method. It’s super quick and easy. I also felt that it helped me simplify fractions!”
 “If I had learned this method in High School, I would have passed [math] in High School!”
 “It’s simple and clean when written out so I don’t get confused by so many things going on at once.”
 “My favorite method is the [Synthetic] factoring method. I find it faster for me. Also, I was able to memorize the steps quickly.”
 “I love the [Synthetic Factoring] way. It’s simpler to me; my brain recognizes it. I love plugging in the answers!”

STUDENT THOUGHTS ON
SYNTHETIC FACTORING
• Another survey was performed in Fall 2011 of 46 students.
• This survey measured the following questions:
1.
Synthetic factoring has increased the speed of which I can factor trinomials.
2.
Synthetic factoring has increased the accuracy of which I can factor trinomials.
3.
Synthetic Factoring is easy to learn.
4.
I was able to grasp and understand the synthetic factoring method the first time it was shown to me.
5.
My confidence for factoring trinomials has increased since learning synthetic factoring.
6.
I like the fact that I only have to learn one technique for factoring any type of trinomial.
7.
I feel that synthetic factoring is easier to remember than other methods I have learned.

STUDENT THOUGHTS ON
SYNTHETIC FACTORING
1. Speed
2. Accuracy
3. Easy to learn
4. Learn first time
5. Confidence
6. Only one method
7. Easy to remember
75.6%
62.6%
75.6%
60.0%
55.6%
71.1%
70.5%
24.4%
37.8%
22.2%
37.8%
44.4%
29.5%
29.5%
0%
0%
2.2%
2.2%
0%
2.2%
0%
0%
0%
0%
0%
0%
0%
0%

STUDENT THOUGHTS ON
SYNTHETIC FACTORING
• Comments from the Fall 2011 survey:
 “This method is amazing and has helped me to like factoring a lot more than I used to.”
 “I have not learned any other form that I can remember. But, using the
Synthetic Factoring is really useful and helpful to me. I used this method in
[my Beginning Algebra course] and still remembered it a year later. I love this method!”
 “This method is great because prior to this, I was unable to factor most of the time. This allows me to get it right every time the first time.”
 “I use [Synthetic factoring] every time since I learned [it]!”
 “Before, factoring with a coefficient besides 1 in front of x made no sense, but synthetic factoring immediately clicked.”
 “It’s hard to learn and to get if you don’t know your multiplication tables.”

FACULTY RESPONSE
• From the beginning, the thing that has most impressed us is the speed with which we can teach factoring.
• It only takes one day to teach factoring trinomials, regardless of form.
• Have seen students’ tests scores increase because of Synthetic Factoring over the last year and a half.
• Several other faculty members have used this method in their courses, adding it along side the other methods presented in the books.
• This is being considered for inclusion into Beginning Algebra,
Intermediate Algebra, and Beginning and Intermediate Algebra text books.

USING SYNTHETIC FACTORING TO
SOLVE QUADRATIC EQUATIONS
• In the Summer of 2010, while teaching Synthetic Factoring, we discovered that the reduced fractions found using this method are the additive inverses of the zeros of the polynomial.
• For instance, solve: 5𝑥 2 − 12𝑥 + 4 = 0
1.
5 ∙ 4 = 20
2.
−10
5
−2
1
,
−2
5
3.
𝑥 − 2 5𝑥 − 2 = 0
4.
𝑥 − 2 = 0, 5𝑥 − 2 = 0
2
5.
𝑥 = 2, 𝑥 =
5
• Notice that we can skip writing out the factors and solving both using the zero product rule because the answers (2 and
2 fractions (-2 and factoring is used.
5
) are the additive inverses of our
−2
5
)! Therefore, we can skip steps 3 and 4 when synthetic

FURTHER IMPLICATIONS OF
SYNTHETIC FACTORING
• Synthetic Factoring is a quick method for finding the zeros of a factorable trinomial equation.
• Synthetic Factoring is directly related to the Rational Zeros Theorem where all rational zeros can be written as 𝑎 𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 𝑎
0 𝑎 𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 𝑎 𝑛 fractions is a single zero of the polynomial.
. Thus, each of the
• Synthetic Factoring reinforces skills that the students have already learned, such as simplifying fractions.
• The hardest part of factoring is now finding the factors of 𝑎𝑐 that add to 𝑏 .

PROOF OF SYNTHETIC FACTORING
• Setup: Factoring the Second-Degree polynomial 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 where
1.
𝑎, 𝑏, 𝑐 ∈ ℤ
2.
𝑎 > 0
3.
𝑎, 𝑏, 𝑐 are relatively prime
• Suppose 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = (𝑝𝑥 + 𝑞)(𝑟𝑥 + 𝑠)
• For uniqueness, 𝑝, 𝑞, 𝑟, 𝑠 ∈ ℤ and 𝑝 > 0, 𝑟 > 0
• Because 𝑎, 𝑏, 𝑐 are relatively prime, the pairs (𝑝, 𝑞) and (𝑟, 𝑠) must also be relatively prime

PROOF OF SYNTHETIC FACTORING
• Simplifying gives:
• 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
• 𝑝𝑥 + 𝑞 𝑟𝑥 + 𝑠
• 𝑝𝑟𝑥 2 + 𝑝𝑠𝑥 + 𝑞𝑟𝑥 + 𝑞𝑠
• 𝑝𝑟 𝑥 2 + 𝑝𝑠 + 𝑞𝑟 𝑥 + (𝑞𝑠)
• Thus, 𝑎 = 𝑝𝑟 , 𝑏 = 𝑝𝑠 + 𝑞𝑟 , and 𝑐 = 𝑞𝑠
• Let 𝑚 = 𝑝𝑠 and 𝑛 = 𝑞𝑟 such that 𝑚𝑛 = 𝑝𝑞𝑟𝑠 = 𝑎𝑐 and 𝑚 + 𝑛 = 𝑝𝑠 + 𝑞𝑟 = 𝑏

PROOF OF SYNTHETIC FACTORING
• Method:
• Assume there exists some 𝑚, 𝑛 such that 𝑚 𝑎
= 𝑝𝑠 𝑝𝑟
= 𝑟 𝑠 and 𝑛 𝑎
= 𝑞𝑟 𝑝𝑟
= 𝑞 𝑝
• Because 𝑠, 𝑟 and 𝑝, 𝑞 are already relatively prime, simplified fraction equivalent to 𝑚 𝑎 𝑠 is a completely 𝑟 is a completely simplified fraction equivalent to 𝑛 and 𝑞 𝑝 𝑎
.
• This makes 𝑝, 𝑞, 𝑟, 𝑠 uniquely separate from 𝑎, 𝑏, 𝑐, 𝑚, 𝑛 .
• Since 𝑝, 𝑞, 𝑟, 𝑠 ∈ ℤ and 𝑝 > 0, 𝑟 > 0 , 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = (𝑝𝑥 + 𝑞)(𝑟𝑥 + 𝑠)

QUESTIONS?
• Contact us at:
• Darren Wiberg: wibergdb@uvu.edu
• Jonathan Anderson: jonathana@uvu.edu