lesson05.pdf

THE UNIVERSITY OF AKRON

Mathematics and Computer Science

mptii

Lesson 5: Expansion

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IamD S

N ⊆ Z ⊆ Q ⊆ R ⊆ C

a 3 a 4 = a 7 ( ab ) 10 = a 10 b 10

− ( ab − (3 ab − 4))=2 ab − 4

( ab ) 3 ( a − 1 + b − 1 )=( ab ) 2 ( a + b )

( a − b ) 3 = a 3 − 3 a 2 b +3 ab 2 − b 3

2 x 2 − 3 x − 2=(2 x + 1)( x − 2)

2 x +13=0 = ⇒ x = − 26

G= { ( x, y ) | y = f ( x ) }

f ( x )= mx + b

y = sin x

Last Revision Date: 8/19/2000

Lesson 5: Expansion

Table of Contents

5. Expansion

5.1. General Methods of Expansion

• Special Products • Radicals Revisited

5.2. The Binomial Formula

• The Binomial Expansion Algorithm

5. Expansion

In this lesson we discuss and illustrate methods of expanding a prod-

uct of algebraic expressions. Initially, we discuss General Methods

of Expansion . Following that, we consider the problem of computing

powers of a binomial. Such expansion can be carried out quicklyand

quietlybyusing the Binomial Formula .

The student will ﬁnd these techniques to be invaluable tools in their

studyof algebra, calculus, and beyond ... perhaps into an engineering

discipline.

5.1. General Methods of Expansion

In Lesson 4 , in the section entitled The Distributive Law ,we

looked at products of the form a ( b + c ) and saw that

a ( b + c )= ab + ac.

(1)

When read from left-to-right, the formula is a rule for expanding a

product. When read from right-to-left, the above formula can be used

for simple factoring or combining of similar terms.

Section 5: Expansion

In this lesson we take up the problem of expanding more complicated

expressions than we considered in Lesson 4 . These more complicated

products are products in which both factors are the sum of several

terms: such as ( x 2 +2)(3 x− 4), ( a + b )( a + b + c ), or ( x +1) 2 ( x +2) 3 .

An algebraic expression consisting of exactlytwo terms is called a

binomial . Consider the problem of computing the product of two bi-

nomials: ( a + b )( c + d ). We can and do expand this product bythe

Distributive Law .

( a + b )( c + d )= a · ( c + d )+ b · ( c + d )

by (1)

= ac + ad + bc + bd

by (1)

Thus,

General Multiplication Rule :

( a + b )( c + d )= ac + ad + bc + bd

(2)

Section 5: Expansion

KeyPoint. If you study the formula (2) , you can see that the product

of two binomials is the sum of all possible products obtained bytaking

one term from the ﬁrst factor and one term from the second factor.

This observation is valid even when there are an arbitrarynumber

of factors in the product and an arbitrarynumber of terms in each

factor.

The above observation then eliminates the need to memorize for-

mula (2) ! Let’s go the examples.

Example 5.1. Expand and combine each of the products.

(a) ( x + 1)( x + 2)

(b) (2 w − 3 s )(5 w +2 s )

Studythe above examples. Tryto “see” the pattern and get a “feel”

for multiplying out binomials.

Exercise 5.1. Expand and combine each of the following.

(a) (4 x − 5)(3 x + 2) (b) ( x − 2 y ) 2

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