<< Chapter < Page Chapter >> Page >

Access these online resources for additional instruction and practice with special products:

Key concepts

  • Binomial Squares Pattern
    • If a , b are real numbers,
      No Alt Text
    • ( a + b ) 2 = a 2 + 2 a b + b 2
    • ( a b ) 2 = a 2 2 a b + b 2
    • To square a binomial: square the first term, square the last term, double their product.

  • Product of Conjugates Pattern
    • If a , b are real numbers,
      No Alt Text
    • ( a b ) ( a + b ) = a 2 b 2
    • The product is called a difference of squares.

  • To multiply conjugates:
    • square the first term square the last term write it as a difference of squares

Practice makes perfect

Square a Binomial Using the Binomial Squares Pattern

In the following exercises, square each binomial using the Binomial Squares Pattern.

( q + 12 ) 2

q 2 + 24 q + 144

Got questions? Get instant answers now!

( x + 2 3 ) 2

x 2 + 4 3 x + 4 9

Got questions? Get instant answers now!

( y 6 ) 2

y 2 12 y + 36

Got questions? Get instant answers now!

( p 13 ) 2

p 2 26 p + 169

Got questions? Get instant answers now!

( 4 a + 10 ) 2

16 a 2 + 80 a + 100

Got questions? Get instant answers now!

( 3 z + 1 5 ) 2

9 z 2 + 6 5 z + 1 25

Got questions? Get instant answers now!

( 2 y 3 z ) 2

4 y 2 12 y z + 9 z 2

Got questions? Get instant answers now!

( 1 8 x 1 9 y ) 2

1 64 x 2 1 36 x y + 1 81 y 2

Got questions? Get instant answers now!

( 5 u 2 + 9 ) 2

25 u 4 + 90 u 2 + 81

Got questions? Get instant answers now!

( 8 p 3 3 ) 2

64 p 6 48 p 3 + 9

Got questions? Get instant answers now!

Multiply Conjugates Using the Product of Conjugates Pattern

In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.

( c 5 ) ( c + 5 )

c 2 25

Got questions? Get instant answers now!

( x + 3 4 ) ( x 3 4 )

Got questions? Get instant answers now!

( b + 6 7 ) ( b 6 7 )

b 2 36 49

Got questions? Get instant answers now!

( 5 k + 6 ) ( 5 k 6 )

Got questions? Get instant answers now!

( 8 j + 4 ) ( 8 j 4 )

64 j 2 16

Got questions? Get instant answers now!

( 11 k + 4 ) ( 11 k 4 )

Got questions? Get instant answers now!

( 9 c + 5 ) ( 9 c 5 )

81 c 2 25

Got questions? Get instant answers now!

( 13 q ) ( 13 + q )

169 q 2

Got questions? Get instant answers now!

( 5 3 x ) ( 5 + 3 x )

Got questions? Get instant answers now!

( 4 6 y ) ( 4 + 6 y )

16 36 y 2

Got questions? Get instant answers now!

( 9 c 2 d ) ( 9 c + 2 d )

Got questions? Get instant answers now!

( 7 w + 10 x ) ( 7 w 10 x )

49 w 2 100 x 2

Got questions? Get instant answers now!

( m + 2 3 n ) ( m 2 3 n )

Got questions? Get instant answers now!

( p + 4 5 q ) ( p 4 5 q )

p 2 16 25 q 2

Got questions? Get instant answers now!

( a b 4 ) ( a b + 4 )

Got questions? Get instant answers now!

( x y 9 ) ( x y + 9 )

x 2 y 2 81

Got questions? Get instant answers now!

( u v 3 5 ) ( u v + 3 5 )

Got questions? Get instant answers now!

( r s 2 7 ) ( r s + 2 7 )

r 2 s 2 4 49

Got questions? Get instant answers now!

( 2 x 2 3 y 4 ) ( 2 x 2 + 3 y 4 )

Got questions? Get instant answers now!

( 6 m 3 4 n 5 ) ( 6 m 3 + 4 n 5 )

36 m 6 16 n 10

Got questions? Get instant answers now!

( 12 p 3 11 q 2 ) ( 12 p 3 + 11 q 2 )

Got questions? Get instant answers now!

( 15 m 2 8 n 4 ) ( 15 m 2 + 8 n 4 )

225 m 4 64 n 8

Got questions? Get instant answers now!

Recognize and Use the Appropriate Special Product Pattern

In the following exercises, find each product.


( p 3 ) ( p + 3 )
( t 9 ) 2
( m + n ) 2
( 2 x + y ) ( x 2 y )

Got questions? Get instant answers now!
  1. ( 2 r + 12 ) 2
  2. ( 3 p + 8 ) ( 3 p 8 )
  3. ( 7 a + b ) ( a 7 b )
  4. ( k 6 ) 2

4 r 2 + 48 r + 144 9 p 2 64 7 a 2 48 a b 7 b 2 k 2 12 k + 36

Got questions? Get instant answers now!
  1. ( a 5 7 b ) 2
  2. ( x 2 + 8 y ) ( 8 x y 2 )
  3. ( r 6 + s 6 ) ( r 6 s 6 )
  4. ( y 4 + 2 z ) 2
Got questions? Get instant answers now!
  1. ( x 5 + y 5 ) ( x 5 y 5 )
  2. ( m 3 8 n ) 2
  3. ( 9 p + 8 q ) 2
  4. ( r 2 s 3 ) ( r 3 + s 2 )

x 10 y 10 m 6 16 m 3 n + 64 n 2 81 p 2 + 144 p q + 64 q 2 r 5 + r 2 s 2 r 3 s 3 s 5

Got questions? Get instant answers now!

Everyday math

Mental math You can use the product of conjugates pattern to multiply numbers without a calculator. Say you need to multiply 47 times 53. Think of 47 as 50 3 and 53 as 50 + 3 .

  1. Multiply ( 50 3 ) ( 50 + 3 ) by using the product of conjugates pattern, ( a b ) ( a + b ) = a 2 b 2 .
  2. Multiply 47 · 53 without using a calculator.
  3. Which way is easier for you? Why?
Got questions? Get instant answers now!

Mental math You can use the binomial squares pattern to multiply numbers without a calculator. Say you need to square 65. Think of 65 as 60 + 5 .

  1. Multiply ( 60 + 5 ) 2 by using the binomial squares pattern, ( a + b ) 2 = a 2 + 2 a b + b 2 .
  2. Square 65 without using a calculator.
  3. Which way is easier for you? Why?

4,225 4,225 Answers will vary.

Got questions? Get instant answers now!

Writing exercises

How do you decide which pattern to use?

Got questions? Get instant answers now!

Why does ( a + b ) 2 result in a trinomial, but ( a b ) ( a + b ) result in a binomial?

Answers will vary.

Got questions? Get instant answers now!

Marta did the following work on her homework paper:

( 3 y ) 2 3 2 y 2 9 y 2

Explain what is wrong with Marta’s work.

Got questions? Get instant answers now!

Use the order of operations to show that ( 3 + 5 ) 2 is 64, and then use that numerical example to explain why ( a + b ) 2 a 2 + b 2 .

Answers will vary.

Got questions? Get instant answers now!

Self check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This is a table that has four rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “square a binomial using the binomial squares pattern,” “multiply conjugates using the product of conjugates pattern,” and “recognize and use the appropriate special product pattern.” The rest of the cells are blank.

On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

Questions & Answers

what is mutation
Janga Reply
what is a cell
Sifune Reply
how is urine form
Sifune
what is antagonism?
mahase Reply
classification of plants, gymnosperm features.
Linsy Reply
what is the features of gymnosperm
Linsy
how many types of solid did we have
Samuel Reply
what is an ionic bond
Samuel
What is Atoms
Daprince Reply
what is fallopian tube
Merolyn
what is bladder
Merolyn
what's bulbourethral gland
Eduek Reply
urine is formed in the nephron of the renal medulla in the kidney. It starts from filtration, then selective reabsorption and finally secretion
onuoha Reply
State the evolution relation and relevance between endoplasmic reticulum and cytoskeleton as it relates to cell.
Jeremiah
what is heart
Konadu Reply
how is urine formed in human
Konadu
how is urine formed in human
Rahma
what is the diference between a cavity and a canal
Pelagie Reply
what is the causative agent of malaria
Diamond
malaria is caused by an insect called mosquito.
Naomi
Malaria is cause by female anopheles mosquito
Isaac
Malaria is caused by plasmodium Female anopheles mosquitoe is d carrier
Olalekan
a canal is more needed in a root but a cavity is a bad effect
Commander
what are pathogens
Don Reply
In biology, a pathogen (Greek: πάθος pathos "suffering", "passion" and -γενής -genēs "producer of") in the oldest and broadest sense, is anything that can produce disease. A pathogen may also be referred to as an infectious agent, or simply a germ. The term pathogen came into use in the 1880s.[1][2
Zainab
A virus
Commander
Definition of respiration
Muhsin Reply
respiration is the process in which we breath in oxygen and breath out carbon dioxide
Achor
how are lungs work
Commander
where does digestion begins
Achiri Reply
in the mouth
EZEKIEL
what are the functions of follicle stimulating harmones?
Rashima Reply
stimulates the follicle to release the mature ovum into the oviduct
Davonte
what are the functions of Endocrine and pituitary gland
Chinaza
endocrine secrete hormone and regulate body process
Achor
while pituitary gland is an example of endocrine system and it's found in the Brain
Achor
what's biology?
Egbodo Reply
Biology is the study of living organisms, divided into many specialized field that cover their morphology, physiology,anatomy, behaviour,origin and distribution.
Lisah
biology is the study of life.
Alfreda
Biology is the study of how living organisms live and survive in a specific environment
Sifune
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 1

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

Ask