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

Key concepts

  • Binomial Squares Pattern
    • If a , b are real numbers,
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    • ( 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,
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    • ( 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

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( x + 2 3 ) 2

x 2 + 4 3 x + 4 9

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( y 6 ) 2

y 2 12 y + 36

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( p 13 ) 2

p 2 26 p + 169

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( 4 a + 10 ) 2

16 a 2 + 80 a + 100

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( 3 z + 1 5 ) 2

9 z 2 + 6 5 z + 1 25

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( 2 y 3 z ) 2

4 y 2 12 y z + 9 z 2

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( 1 8 x 1 9 y ) 2

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

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( 5 u 2 + 9 ) 2

25 u 4 + 90 u 2 + 81

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( 8 p 3 3 ) 2

64 p 6 48 p 3 + 9

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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

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( x + 3 4 ) ( x 3 4 )

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( b + 6 7 ) ( b 6 7 )

b 2 36 49

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( 5 k + 6 ) ( 5 k 6 )

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( 8 j + 4 ) ( 8 j 4 )

64 j 2 16

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( 11 k + 4 ) ( 11 k 4 )

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( 9 c + 5 ) ( 9 c 5 )

81 c 2 25

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( 13 q ) ( 13 + q )

169 q 2

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( 5 3 x ) ( 5 + 3 x )

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( 4 6 y ) ( 4 + 6 y )

16 36 y 2

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

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

49 w 2 100 x 2

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

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

p 2 16 25 q 2

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( a b 4 ) ( a b + 4 )

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( x y 9 ) ( x y + 9 )

x 2 y 2 81

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

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

r 2 s 2 4 49

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

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

36 m 6 16 n 10

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

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

225 m 4 64 n 8

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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 )

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  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

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  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
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  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

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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?
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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.

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Writing exercises

How do you decide which pattern to use?

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

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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.

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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.

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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?

Practice Key Terms 1

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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