Transcript The Factoring Flowchart (pptx)

FACTORING POLYNOMIALS WITH
“THE FACTORING FLOWCHART”
By Kyle Muldrow
Overview
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General Description
Review of Flowcharts and Flowchart Symbols
The Factoring Flowchart
Greatest Common Factor
Factoring Polynomials with two terms
Factoring Polynomials with four terms
Factoring Polynomials with three terms
 Definition of “The Blanks Method”
 Definition of “The Modified Blanks Method”
 Interactive Session
General Description
 In Beginning Algebra, Intermediate Algebra,
and College Algebra courses, one topic that is
covered is factoring polynomials with two,
three, or four terms.
 Many factoring techniques are discussed.
 Factoring by grouping
 Perfect Square
 Difference of Squares
 Sum/Difference of Cubes
General Description (cont’d)
 However, students don’t only need to know the
techniques – they must also know when to use
each technique.
 Textbooks might have suggestions, but they are
usually limited to a list that references sections
of the book.
 “The Factoring Flowchart” combines elements of
a programming flowchart along with new
terminology to give students and teachers a
diagram to follow to find the right fact0ring
technique.
Review of Flowcharts
 A flowchart is “a diagram that graphically
depicts steps that take place in a computer
program.” [1]
 Commonly used symbols are the terminal
symbol (oval), the processing symbol
(rectangle), and the decision symbol
(diamond).
1. Tony Gaddis, Programming Logic and Design, 2nd Edition, pgs. 32, Pearson Education, Inc. (publishing as
Addison-Wesley), New York City, NY, ISBN 978-0-13-607333-2 (2010)
Flowchart Symbols
 Terminal Symbol
 Indicates the Start and End of a program [1]
 Processing Symbol
 Indicates processing (math calculations) done in a
program [1]
 Decision Symbol
 Indicates a condition that must be tested [1]
 All symbols are connected by arrows to
represent the “flow” of the program. [1]
1. Tony Gaddis, Programming Logic and Design, 2nd Edition, pgs. 32, 126-127, Pearson Education, Inc.
(publishing as Addison-Wesley), New York City, NY, ISBN 978-0-13-607333-2 (2010)
Greatest Common Factor (GCF)
 The first thing the Factoring Flowchart says
to do is the first thing you do when factoring
any polynomial: Look for a Greatest Common
Factor in the equation.
 Examples:
 5𝑥 5 − 25𝑥 3 + 40𝑥 2
 GCF = 5𝑥 2 ; Answer is: 5𝑥 2 𝑥 3 − 5𝑥 + 8
 17𝑥 5 𝑦 7 − 68𝑥 10 𝑦 4 + 170𝑥 9 𝑦 8
 GCF = 17𝑥 5 𝑦 4 ;
Answer is:17𝑥 5 𝑦 4 𝑦 3 − 4𝑥 5 + 10𝑥 4 𝑦 4
Factoring Polynomials with
two terms
 After factoring out the Greatest Common Factor
(if any), we now look at how many terms are in
the polynomial.
 If the polynomial has two terms, there are three
possible ways to factor it:
 Difference of Squares: 𝑎2 − 𝑏 2 = (𝑎 + 𝑏)(𝑎 − 𝑏)[2]
 Sum of Cubes: 𝑎3 + 𝑏 3 = (𝑎 + 𝑏)(𝑎2 − 2𝑎𝑏 + 𝑏 2 ) [2]
 Difference of Cubes: 𝑎3 − 𝑏 3 = (𝑎 − 𝑏)(𝑎2 + 2𝑎𝑏 +
𝑏 2 ) [2]
2. Margaret L. Lial, John Hornsby, and Terry McGinnis, Beginning Algebra, 12th Edition, pgs. 423 and 426, Pearson Education,
Inc., New York City, NY, ISBN 978-0-321-96933-0 (2012)
Difference of Squares and
Sum of Cubes
 144𝑥 2 − 169
 Minus sign in middle; both terms perfect squares,
so use Difference of Squares.
 144𝑥 2 − 169 = 12𝑥 2 − 132 = (12𝑥 +
13)(12𝑥 − 13)
 8𝑥 3 + 27
 Plus sign in middle; both terms perfect cubes, so
use sum of cubes.
 8𝑥 3 + 27 = 2𝑥 3 + 33 = (2𝑥 + 3)( 2𝑥 2 −
2𝑥 3 + 32 ) = (2𝑥 + 3)(4𝑥 2 − 6𝑥 + 9)
Difference of Cubes
 27𝑥 3 − 64
 Two terms, minus sign in middle, both terms
perfect cubes, so use difference of cubes.
 27𝑥 3 − 64 = 3𝑥 3 − 43 = (3𝑥 − 4)( 3𝑥 2 +
3𝑥 4 + 42 = (3𝑥 − 4)(9𝑥 2 + 12𝑥 + 16)
Factoring polynomials with
four terms (Grouping)
 If a polynomial has four terms, there is only one
factoring technique that can be used: Factoring
by Grouping
 Example: 7𝑥 3 − 14𝑥 2 𝑦 − 3x𝑦 2 + 6𝑦 3
 1) Pair up terms: (7𝑥 3 − 14𝑥 2 𝑦)(−3x𝑦 2 + 6𝑦 3)
 2) Factor GCF out of each pair:
7𝑥 2 𝑥 − 2𝑦 − 3𝑦 2 𝑥 − 2𝑦
 3) If we have same equation in both parentheses,
we did it right!
Answer: (7𝑥 2 − 3𝑦 2 )(𝑥 − 2𝑦)
Factoring polynomials with
three terms
 If a polynomial has three terms, there are
three possible factoring techniques that can
be used:
 Perfect square
 𝑎2 + 2𝑎𝑏 + 𝑏 2 = 𝑎 + 𝑏 2 or
 𝑎2 − 2𝑎𝑏 + 𝑏 2 = (𝑎 − 𝑏)2 [2]
 The “Blanks” Method
 The “Modified Blanks” Method
2. Margaret L. Lial, John Hornsby, and Terry McGinnis, Beginning Algebra, 12th Edition, pgs. 423 and 426, Pearson Education, Inc.,
New York City, NY, ISBN 978-0-321-96933-0 (2012)
Perfect Square
 3𝑥 2 − 18𝑥𝑦 + 27𝑦 2
 GCF = 3
 3𝑥 2 − 18𝑥𝑦 + 27𝑦 2 = 3(𝑥 2 − 6𝑥𝑦 + 9𝑦 2 )
 First term (𝑥 2 ) and last term 9𝑦 2 are perfect
squares.
 3(𝑥 2 − 6𝑥𝑦 + 9𝑦 2 ) = 3(𝑥 2 − 2(𝑥)(3𝑦) + 3𝑦 2 )
= 3 𝑥 − 3𝑦 2
The “Blanks” Method
 The “Blanks” Method is a factoring technique
that can be used to factor a polynomial that
meets all of the following conditions:
 The polynomial has three terms
 The polynomial has 1 as its leading coefficient
 First term is 𝑥 2 (no number attached to it)
 The polynomial is not a perfect square
 This method can still be used on a perfect square
rather than the perfect square formula seen earlier.
The “Blanks” Method (cont’d)
 𝑥 2 − 7𝑥 − 30
 Three terms, not a perfect square, leading
coefficient is 1.
 When factored, solution will look something like
this: 𝑥 2 − 7𝑥 − 30 = 𝑥 _____ 𝑥 _____
 The numbers that fill in the blanks must meet the
following criteria:
 _____ + _____ = -7 (number attached to x)
 _____ x _____ = -30 (number with no x’s)
The “Blanks” Method (cont’d)
 To determine which numbers go in the
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blanks, look at all possible factorizations of
the number with no x’s.
30 = 1 x 30, 2 x 15, 3 x 10, and 5 x 6
The most likely candidate is 3 x 10. Negative
signs can be added.
3 + (-10) = -7 and 3 x (-10) = -30
𝑥 2 − 7𝑥 − 30 = 𝑥 + 3 𝑥 − 10
The “Modified Blanks” Method
 The “Modified Blanks” Method is a factoring
technique that can be used to factor a
polynomial that meets all of the following
conditions:
 The polynomial has three terms
 The polynomial does not have 1 as its leading
coefficient
 The polynomial is not a perfect square
 This method can still be used on a perfect square
rather than the perfect square formula seen earlier.
“Modified Blanks” (cont’d)
 6𝑥 2 − 11𝑥 + 4
 Three terms, not a perfect square, leading
coefficient is not 1.
 Set up blanks as in the “Blanks” method, with one
change:
 __ x __ = (number with no x’s)(number attached to 𝑥 2 )
 ___ + ___ = -11
 ___ x ___ = 24 (6x4)
 In this case, the numbers -3 and -8 satisfy both
blanks.
“Modified Blanks” (cont’d)
 Now rewrite the original equation with four
terms using the numbers from the blanks:
 6𝑥 2 − 11𝑥 + 4 = 6𝑥 2 − 3𝑥 − 8𝑥 + 4
 Since we now have four terms, use Factoring
by Grouping.
 6𝑥 2 − 3𝑥 − 8𝑥 + 4 = 3x 2𝑥 − 1 − 4 2𝑥 − 1
 Answer: (3x − 4) 2𝑥 − 1
Summary
 The Factoring Flowchart can be used to determine
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the proper factoring technique to use when factoring
polynomials with two, three, or four terms.
The first step is to check if a Greatest Common
Factor can be factored out of all the terms.
Polynomials with two terms can be factored using
Difference of Squares, Sum of Cubes, and Difference
of Cubes.
Polynomials with three terms can be factored by
using Perfect Square, The “Blanks” Method, and The
“Modified Blanks” method.
Polynomials with four terms can be factored by
using Factoring by Grouping.
References
1. Tony Gaddis, Programming Logic and Design,
2nd Edition, pgs. 32, 126-127, Pearson
Education, Inc. (publishing as AddisonWesley), New York City, NY, ISBN 978-0-13607333-2 (2010)
2. Margaret L. Lial, John Hornsby, and Terry
McGinnis, Beginning Algebra, 12th Edition,
pgs. 423 and 426, Pearson Education, Inc.,
New York City, NY, ISBN 978-0-321-96933-0
(2012)