Transcript Synthetic Factoring Powerpoint

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
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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:
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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
𝑛
and 𝑎 .
4. Simplify the fractions formed in step 3 as
5. Form the factors 𝑝𝑥 + 𝑞 𝑟𝑥 + 𝑠 .
6. Check the answer by FOILing the result
𝑚
𝑎
=
𝑞
𝑝
and
𝑛
𝑎
𝑠
𝑟
= .
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
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Another survey was performed in Fall 2011 of 46 students.
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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
Question
Strongly
Agree
Agree
Disagree
Strongly
Disagree
1. Speed
75.6%
24.4%
0%
0%
2. Accuracy
62.6%
37.8%
0%
0%
3. Easy to learn
75.6%
22.2%
2.2%
0%
4. Learn first time
60.0%
37.8%
2.2%
0%
5. Confidence
55.6%
44.4%
0%
0%
6. Only one
method
71.1%
29.5%
2.2%
0%
7. Easy to
remember
70.5%
29.5%
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
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From the beginning, the thing that has most impressed us is the speed
with which we can teach factoring.
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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 −2 −2
,
5
1
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
2
product rule because the answers (2 and ) are the additive inverses of our
5
−2
fractions (-2 and )! Therefore, we can skip steps 3 and 4 when synthetic
5
factoring is used.
FURTHER IMPLICATIONS OF
SYNTHETIC FACTORING
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Synthetic Factoring is a quick method for finding the zeros of a factorable
trinomial equation.
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Synthetic Factoring is directly related to the Rational Zeros Theorem
𝑎 𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 𝑎
where all rational zeros can be written as 𝑎 𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 𝑎 0 . Thus, each of the
fractions is a single zero of the polynomial.
𝑛
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Synthetic Factoring reinforces skills that the students have already
learned, such as simplifying fractions.
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The hardest part of factoring is now finding the factors of 𝑎𝑐 that add to 𝑏.
PROOF OF SYNTHETIC FACTORING
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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
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Suppose 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = (𝑝𝑥 + 𝑞)(𝑟𝑥 + 𝑠)
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For uniqueness, 𝑝, 𝑞, 𝑟, 𝑠 ∈ ℤ and 𝑝 > 0, 𝑟 > 0
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Because 𝑎, 𝑏, 𝑐 are relatively prime, the pairs (𝑝, 𝑞) and (𝑟, 𝑠) must also be
relatively prime
PROOF OF SYNTHETIC FACTORING
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Simplifying gives:
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𝑎𝑥 2 + 𝑏𝑥 + 𝑐
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𝑝𝑥 + 𝑞 𝑟𝑥 + 𝑠
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𝑝𝑟𝑥 2 + 𝑝𝑠𝑥 + 𝑞𝑟𝑥 + 𝑞𝑠
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𝑝𝑟 𝑥 2 + 𝑝𝑠 + 𝑞𝑟 𝑥 + (𝑞𝑠)
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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
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Method:
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Assume there exists some 𝑚, 𝑛 such that
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Because 𝑠, 𝑟 and 𝑝, 𝑞 are already relatively prime,
simplified fraction equivalent to
𝑛
𝑎
𝑚
𝑎
𝑞
𝑚
𝑎
=
𝑝𝑠
𝑝𝑟
𝑠
𝑟
= and
𝑠
𝑟
𝑛
𝑎
=
𝑞𝑟
𝑝𝑟
=
𝑞
𝑝
is a completely
and 𝑝 is a completely simplified
fraction equivalent to .
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This makes 𝑝, 𝑞, 𝑟, 𝑠 uniquely separate from 𝑎, 𝑏, 𝑐, 𝑚, 𝑛.
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Since 𝑝, 𝑞, 𝑟, 𝑠 ∈ ℤ and 𝑝 > 0, 𝑟 > 0, 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = (𝑝𝑥 + 𝑞)(𝑟𝑥 + 𝑠)
QUESTIONS?
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Contact us at:
•
Darren Wiberg: [email protected]
•
Jonathan Anderson: [email protected]