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Factor by Grouping

Site: Harrison
Course: Michigan Algebra I Sept. 2012
Book: Factor by Grouping
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Date: Thursday, 21 November 2024, 9:13 PM

Description

Factor by Grouping

Table of contents

Introduction

Another way to solve polynomial equations is to factor. Similar to what was presented in the quadratics unit. There are multiple ways to factor polynomials. In a past unit, many trinomial quadratic equations were able to be factored. However, not all polynomials can be factored in this way. There might be more than three terms or the polynomial may not be made of perfect squares. When this happens, sometimes a common factor can be pulled out. Sometimes grouping the terms of the polynomial into two groups will help to factor the polynomial. Some of these ways will be discussed in this unit.

If you need review of factoring trinomials, please select the following link:

Factoring Trinomials


Factoring

When a polynomial is written as the product of two or more polynomials, the polynomial has been factored. Here is an example:

x3 - 3x2 - 2x +6 = (x - 3)(x2 - 2)

The polynomials (x - 3) and (x2 - 2) are called factors of the polynomial. Note that the degrees of the factors, 1 and 2, respectively, add up to the degree of the original polynomial. Thus factoring breaks up a complicated polynomial into lower degree pieces. In this case, one of the factors can be factored further:

(x - 3)(x2 - 2) = (x - 3) (x + SquareRoot2 )(x - SquareRoot2 )

Now that the polynomial has been factored into three linear or degree 1 polynomials, the polynomial cannot be factored any farther. Therefore, this polynomial has been factored completely.



Review of GCF

If a reminder is needed of factoring out the greatest common factor or GCF, please select the following links for a video lesson and guided practice:

GCF Video Lesson

GCF Guided Practice

By Grouping

When polynomials have an even number of terms, sometimes they can be factored by grouping. The example below will demonstrate how to factor by grouping.

Example Factor and solve the following polynomial:

FactoringEx1-1

Step 1. Use parentheses to group the terms into equal groups.

FactoringEx1-2

Step 2. Factor the GCF from each group.

FactoringEx1-3

Step 3. Since the two remaining factors are both (x + 3), use Distributive Property to regroup.

FactoringEx1-4

Step 4. Use Zero Product Property to solve the polynomial.

FactoringEx1-5 or FactoringEx1-6

*Note: Since only one of these solutions is a real number, there is only one x -intercept on the graph of the corresponding function.

The Equal Test

It is possible to use a TI 83/84 calculator to check if the factoring is correct. In order to do this, follow these steps:

Step 1. On the home screen, enter the original equation.

Step 2. Push the 2ND key and then the TEST key, then press #1 and ENTER to put an "=" sign in.

Step 3. Put in the factored form after the "=" and press ENTER.

Step 4. If the result is 1, the two expressions are equivalent. You have factored correctly! If the result is 0, the factored form is incorrect.

Listen

Video Lessons

To learn more about factoring by grouping, please select the following links:

Factor by Grouping #1

Factor by Grouping #2

Guided Practice

To solidify your understanding of factoring polynomials, visit the following link to Holt, Rinehart and Winston Homework Help Online. It provides examples, video tutorials and interactive practice with answers available. The Practice and Problem Solving section has two parts. The first part offers practice with a complete video explanation for the type of problem with just a click of the video icon. The second part offers practice with the solution for each problem only a click of the light bulb away.

Guided Practice #1

Guided Practice #2

Practice

Factor by Grouping Worksheet

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Answer Key

Factor by Grouping Answer Key

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Sources

Embracing Mathematics, Assessment & Technology in High Schools; A Michigan Mathematics & Science Partnership Grant Project

Holt, Rinehart & Winston, "Factoring by Grouping." http://my.hrw.com/math06_07/nsmedia/lesson_videos/alg1/player.html?contentSrc=7550/7550.xml (accessed 9/8/2010).

Holt, Rinehart & Winston, "Factoring by Using the GCF." http://my.hrw.com/math06_07/nsmedia/lesson_videos/alg1/player.html?contentSrc=6366/6366.xml (accessed 9/8/2010).