Skip to ContentGo to accessibility pageKeyboard shortcuts menu
OpenStax Logo
Algebra 1

6.2.4 Multiplying Special Products

Algebra 16.2.4 Multiplying Special Products
Search for key terms or text.

Activity

Multiply each expression. Identify each as a binomial square or a conjugate pair (difference of squares).

1. ( 3 x − 7 ) 2 ( 3 x − 7 ) 2

2. ( 6 z + 4 ) ( 6 z − 4 ) ( 6 z + 4 ) ( 6 z − 4 )

3. ( 4 a + 2 b ) ( 4 a + 2 b ) ( 4 a + 2 b ) ( 4 a + 2 b )

4. ( 3 w − 9 p ) ( 3 w + 9 p ) ( 3 w − 9 p ) ( 3 w + 9 p )

5. ( 2 y − 8 n ) ( 2 y + 8 n ) ( 2 y − 8 n ) ( 2 y + 8 n )

6. ( 5 k + 2 ) ( 5 k + 2 ) ( 5 k + 2 ) ( 5 k + 2 )

Self Check

Multiply.

( 3 f − 6 g ) 2

  1. 9 f 2 + 36 g 2
  2. 9 f 2 − 36 f g − 36 g 2
  3. 9 f 2 − 36 f g + 36 g 2
  4. 9 f 2 − 18 f g + 36 g 2

Additional Resources

Multiplying Binomial Squares

Mathematicians like to look for patterns that will make their work easier. A good example of this is squaring binomials. While you can always get the product by writing the binomial twice and multiplying the two binomials, there is less work to do if you learn to use a pattern. Let’s start by looking at three examples and look for a pattern.

Look at these results. Do you see any patterns?

A grid showing three algebraic binomials squared on top, their expanded forms in the middle, and simplified results at the bottom: (x+9)², (y-7)², (2x+3)², with corresponding steps below each.

What about the number of terms? In each example, we squared a binomial, and the result was a trinomial.

( a + b ) 2 = ____ + ____ + ____ ( a + b ) 2 = ____ + ____ + ____

Now look at the first term in each result. Where did it come from?

The first term is the product of the first terms of each binomial. Since the binomials are identical, it is just the square of the first term!

( a + b ) 2 = a 2 + ____ + ____ ( a + b ) 2 = a 2 + ____ + ____

To get the first term of the product, square the first term.

Where did the last term come from? Look at the examples and find the pattern.

The last term is the product of the last terms, which is the square of the last term.

( a + b ) 2 = ____ + ____ + b 2 ( a + b ) 2 = ____ + ____ + b 2

To get the last term of the product, square the last term.

Finally, look at the middle term. Notice it came from adding the “outer” and the “inner” terms—which are the same! So the middle term is double the product of the two terms of the binomial.

( a + b ) 2 = ____ + 2 a b + ____ ( a + b ) 2 = ____ + 2 a b + ____

( a + b ) 2 = ____ + 2 a b + ____ ( a + b ) 2 = ____ + 2 a b + ____

To get the middle term of the product, multiply the terms and double their product.

Putting it all together:

Mathematical explanation of binomial squares: (a+b)² = a² + 2ab + b² and (a-b)² = a²-2ab + b², with terms labeled as binomial, first, last, and product terms. Instructions below clarify the pattern.

Try it

Try It: Multiplying Binomial Squares

1. Multiply ( x + 5 ) 2 ( x + 5 ) 2 .

2. Multiply ( 2 x − 3 y ) 2 (2x−3y ) 2 .

Write down your answer, then select the solution button to compare your work.

Conjugate Pairs

We just saw a pattern for squaring binomials that we can use to make multiplying some binomials easier. Similarly, there is a pattern for another product of binomials.

A pair of binomials that each have the same first term and the same last term, but one is a sum and one is a difference, is called a conjugate pair and is of the form ( a − b ) ( a − b ) , ( a + b ) ( a + b ) .

CONJUGATE PAIR

A conjugate pair is two binomials of the form

( a − b ) , ( a + b ) ( a − b ) , ( a + b )

The pair of binomials each have the same first term and the same last term, but one binomial is a sum and the other is a difference.

There is a nice pattern for finding the product of conjugates. You could, of course, simply FOIL to get the product, but using the pattern makes your work easier. Let’s look for the pattern by using FOIL to multiply some conjugate pairs.

Six algebraic expressions are shown, each with a pair of binomials being multiplied above their expanded quadratic forms: (x+9)(x-9), (y-8)(y+8), (2x-5)(2x+5), and their expanded forms below.

What do you observe about the products?

The product of the two binomials is also a binomial! Most of the products resulting from FOIL have been trinomials.

Each first term is the product of the first terms of the binomials, and since they are identical, it is the square of the first term.

( a + b ) ( a − b ) = a 2 − ____ ( a + b ) ( a − b ) = a 2 − ____

To get the first term, square the first term.

The last term came from multiplying the last terms, and it is the square of the last term.

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

To get the last term, square the last term.

Why is there no middle term? Notice the two middle terms you get from FOIL combine to 0 in every case, the result of one addition and one subtraction.

The product of conjugates is always of the form a 2 − b 2 a 2 − b 2 . This is called a difference of squares.

This leads to the pattern:

PRODUCT OF CONJUGATES PATTERN

If a and b are real numbers,

A diagram explains the product of conjugates: (a - b)(a + b) = a² - b². Arrows label (a - b) and (a + b) as conjugates, and a² - b² as the difference of squares. The product is called a difference of squares. To muliply conjugates, square the first term, square the last term, write it as a difference of squares.

We just developed special product patterns for Binomial Squares and for the Product of Conjugates. The products look similar, so it is important to recognize when it is appropriate to use each of these patterns and to notice how they differ. Look at the two patterns together and note their similarities and differences.

COMPARING THE SPECIAL PRODUCT PATTERNS

Binomial Squares Product of Conjugates

( a + b ) 2 = a 2 + 2 a b + b 2 ( a + b ) 2 = a 2 + 2 a b + b 2

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

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

Squaring a binomial.

Multiplying conjugates.

Product is a trinomial.

Product is a binomial.

Inner and outer terms with FOIL are the same .

Inner and outer terms with FOIL are opposites .

Middle term is double the product of the two terms.

There is no middle term.

Try it

Try It: Conjugate Pairs

1.

1. Multiply ( 2 x + 5 ) ( 2 x − 5 ) ( 2 x + 5 ) ( 2 x − 5 ) .

2.

2. Multiply ( 5 m − 9 n ) ( 5 m + 9 n ) ( 5 m − 9 n ) ( 5 m + 9 n ) .

PreviousNext
Citation/Attribution

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution-NonCommercial-ShareAlike License and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:

    Access for free at https://openstax.org/books/algebra-1/pages/about-this-course

  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:

    Access for free at https://openstax.org/books/algebra-1/pages/about-this-course

Citation information

© May 21, 2025 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.