Jon Anderson Darren Wiberg

1. Synthetic Factoring: a Student centered approach in factoring trinomials Jon Anderson Darren Wiberg

2. 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

3. 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

4. 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

5. Introducing synthetic factoring • So, what is Synthetic Factoring? Here are the steps to use: • Given a trinomial in the form of with , , and having no common factors. If , , and have a common factor, factor it out, then begin with step 1 on the resulting trinomial. • Find the value of . • Find a pair of integers and where and . • Create fractions of the form and . • Simplify the fractions formed in step 3 as and . • Form the factors . • Check the answer by FOILing the result

6. 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

7. 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

8. 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!”

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

10. student THOUGHTS ON SYNTHETIC FACTORING

11. 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.”

12. 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.

13. 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: • Notice that we can skip writing out the factors and solving both using the zero product rule because the answers (2 and ) are the additive inverses of our fractions (-2 and )! Therefore, we can skip steps 3 and 4 when synthetic factoring is used.

14. 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 . Thus, each of the fractions is a single zero of the polynomial. • 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 that add to .

15. Proof of synthetic factoring • Setup: Factoring the Second-Degree polynomial where • are relatively prime • Suppose • For uniqueness, and • Because are relatively prime, the pairs and must also be relatively prime

16. Proof of synthetic factoring • Simplifying gives: • Thus, , , and • Let and such that and

17. Proof of synthetic factoring • Method: • Assume there exists some such that and • Because and are already relatively prime, is a completely simplified fraction equivalent to and is a completely simplified fraction equivalent to . • This makes uniquely separate from . • Since and ,

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