FACTORING POLYNOMIALS WITH “THE FACTORING FLOWCHART”

By Kyle Muldrow

Overview

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:

GCF = ; Answer is:

GCF = Answer is:

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: [2] Sum of Cubes: [2] Difference of Cubes: [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

Minus sign in middle; both terms perfect squares, so use Difference of Squares.

Plus sign in middle; both terms perfect cubes, so use sum of cubes.

Difference of Cubes

Two terms, minus sign in middle, both terms perfect cubes, so use difference of cubes.

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: 1) Pair up terms: ) 2) Factor GCF out of each pair:

3) If we have same equation in both parentheses, we did it right!

Answer:

Factoring polynomials with three terms If a polynomial has three terms,

there are three possible factoring techniques that can be used: Perfect square

or [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

GCF = 3

First term and last term are perfect squares.

= =

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 (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)

Three terms, not a perfect square, leading coefficient is 1.

When factored, solution will look something like this:

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 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

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)

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 )

___ + ___ = -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: =

Since we now have four terms, use Factoring by Grouping.

Answer:

Summary

The Factoring Flowchart can be used to determine 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 Addison-Wesley), New York City, NY, ISBN 978-0-13-607333-2 (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)