Learn on PengiOpenstax Elementary Algebra 2EChapter 6: Polynomials

Lesson 4: Special Products

In this lesson from OpenStax Elementary Algebra 2E, students learn to apply the Binomial Squares Pattern and the Product of Conjugates Pattern to multiply special polynomial expressions efficiently. The lesson covers squaring a binomial using the formula (a+b)² = a² + 2ab + b² and multiplying conjugate pairs to produce a difference of squares. Students also practice recognizing which special product pattern applies to a given expression, building fluency with polynomial multiplication techniques used throughout algebra.

Section 1

📘 Special Products

New Concept

This lesson reveals special product patterns as powerful shortcuts for multiplying binomials. You'll master squaring binomials and multiplying conjugates, transforming complex problems into simple, recognizable steps using formulas like $(a+b)^2 = a^2 + 2ab + b^2$ .

What’s next

Get ready for a series of interactive examples and practice cards. You'll apply these special product patterns step-by-step to master the shortcuts.

Section 2

Binomial Squares Pattern

Property

If $a$ and $b$ are real numbers,

(a+b)^2 = a^2 + 2ab + b^2

(a-b)^2 = a^2 - 2ab + b^2

This pattern is a shortcut for squaring a binomial. To use it, you square the first term, square the last term, and then add or subtract double the product of the two terms.

Examples

To multiply $(z+6)^2$ , we use the pattern $(a+b)^2 = a^2 + 2ab + b^2$ . Here, $a=z$ and $b=6$ . The result is $(z)^2 + 2(z)(6) + (6)^2$ , which simplifies to $z^2 + 12z + 36$ .

To multiply $(3y+4)^2$ , we identify $a=3y$ and $b=4$ . Using the pattern, we get $(3y)^2 + 2(3y)(4) + (4)^2$ . This simplifies to $9y^2 + 24y + 16$ .

Section 3

Product of Conjugates Pattern

Property

A conjugate pair is two binomials of the form $(a-b)$ and $(a+b)$ . They have the same first term and the same last term, but one is a sum and the other is a difference.

If $a$ and $b$ are real numbers, the Product of Conjugates Pattern is:

(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, and write the result as a difference.

Examples

To multiply $(p-10)(p+10)$ , we recognize this as a product of conjugates where $a=p$ and $b=10$ . Using the pattern $a^2 - b^2$ , we get $(p)^2 - (10)^2$ , which simplifies to $p^2 - 100$ .

Section 4

Recognizing Special Products

Property

Comparing the Special Product Patterns

Binomial Squares	Product of Conjugates
$(a+b)^2 = a^2 + 2ab + b^2$	$(a-b)(a+b) = a^2 - b^2$
$(a-b)^2 = a^2 - 2ab + b^2$

Squaring a binomial results in a trinomial.
Multiplying conjugates results in a binomial (a difference of squares).
With binomial squares, the inner and outer FOIL terms are the same.
With conjugates, the inner and outer FOIL terms are opposites.

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 6: Polynomials

Back to Openstax Elementary Algebra 2E

Lesson 1
Lesson 1: Add and Subtract Polynomials
Learn Practice
Lesson 2
Lesson 2: Use Multiplication Properties of Exponents
Learn Practice
Lesson 3
Lesson 3: Multiply Polynomials
Learn Practice
Lesson 4Current
Lesson 4: Special Products
Learn Practice
Lesson 5
Lesson 5: Divide Monomials
Learn Practice
Lesson 6
Lesson 6: Divide Polynomials
Learn Practice
Lesson 7
Lesson 7: Integer Exponents and Scientific Notation
Learn Practice

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

📘 Special Products

New Concept

What’s next

Get ready for a series of interactive examples and practice cards. You'll apply these special product patterns step-by-step to master the shortcuts.

Section 2

Binomial Squares Pattern

Property

If $a$ and $b$ are real numbers,

(a+b)^2 = a^2 + 2ab + b^2

(a-b)^2 = a^2 - 2ab + b^2

This pattern is a shortcut for squaring a binomial. To use it, you square the first term, square the last term, and then add or subtract double the product of the two terms.

Examples

To multiply $(z+6)^2$ , we use the pattern $(a+b)^2 = a^2 + 2ab + b^2$ . Here, $a=z$ and $b=6$ . The result is $(z)^2 + 2(z)(6) + (6)^2$ , which simplifies to $z^2 + 12z + 36$ .

To multiply $(3y+4)^2$ , we identify $a=3y$ and $b=4$ . Using the pattern, we get $(3y)^2 + 2(3y)(4) + (4)^2$ . This simplifies to $9y^2 + 24y + 16$ .

Section 3

Product of Conjugates Pattern

Property

A conjugate pair is two binomials of the form $(a-b)$ and $(a+b)$ . They have the same first term and the same last term, but one is a sum and the other is a difference.

If $a$ and $b$ are real numbers, the Product of Conjugates Pattern is:

(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, and write the result as a difference.

Examples

To multiply $(p-10)(p+10)$ , we recognize this as a product of conjugates where $a=p$ and $b=10$ . Using the pattern $a^2 - b^2$ , we get $(p)^2 - (10)^2$ , which simplifies to $p^2 - 100$ .

Section 4

Recognizing Special Products

Property

Comparing the Special Product Patterns

Binomial Squares	Product of Conjugates
$(a+b)^2 = a^2 + 2ab + b^2$	$(a-b)(a+b) = a^2 - b^2$
$(a-b)^2 = a^2 - 2ab + b^2$

Squaring a binomial results in a trinomial.
Multiplying conjugates results in a binomial (a difference of squares).
With binomial squares, the inner and outer FOIL terms are the same.
With conjugates, the inner and outer FOIL terms are opposites.

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 6: Polynomials

Back to Openstax Elementary Algebra 2E

Lesson 1
Lesson 1: Add and Subtract Polynomials
Learn Practice
Lesson 2
Lesson 2: Use Multiplication Properties of Exponents
Learn Practice
Lesson 3
Lesson 3: Multiply Polynomials
Learn Practice
Lesson 4Current
Lesson 4: Special Products
Learn Practice
Lesson 5
Lesson 5: Divide Monomials
Learn Practice
Lesson 6
Lesson 6: Divide Polynomials
Learn Practice
Lesson 7
Lesson 7: Integer Exponents and Scientific Notation
Learn Practice

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

📘 Special Products

New Concept

What’s next

Get ready for a series of interactive examples and practice cards. You'll apply these special product patterns step-by-step to master the shortcuts.

Section 2

Binomial Squares Pattern

Property

If $a$ and $b$ are real numbers,

(a+b)^2 = a^2 + 2ab + b^2

(a-b)^2 = a^2 - 2ab + b^2

This pattern is a shortcut for squaring a binomial. To use it, you square the first term, square the last term, and then add or subtract double the product of the two terms.

Examples

To multiply $(z+6)^2$ , we use the pattern $(a+b)^2 = a^2 + 2ab + b^2$ . Here, $a=z$ and $b=6$ . The result is $(z)^2 + 2(z)(6) + (6)^2$ , which simplifies to $z^2 + 12z + 36$ .

To multiply $(3y+4)^2$ , we identify $a=3y$ and $b=4$ . Using the pattern, we get $(3y)^2 + 2(3y)(4) + (4)^2$ . This simplifies to $9y^2 + 24y + 16$ .

Section 3

Product of Conjugates Pattern

Property

A conjugate pair is two binomials of the form $(a-b)$ and $(a+b)$ . They have the same first term and the same last term, but one is a sum and the other is a difference.

If $a$ and $b$ are real numbers, the Product of Conjugates Pattern is:

(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, and write the result as a difference.

Examples

To multiply $(p-10)(p+10)$ , we recognize this as a product of conjugates where $a=p$ and $b=10$ . Using the pattern $a^2 - b^2$ , we get $(p)^2 - (10)^2$ , which simplifies to $p^2 - 100$ .

Section 4

Recognizing Special Products

Property

Comparing the Special Product Patterns

Binomial Squares	Product of Conjugates
$(a+b)^2 = a^2 + 2ab + b^2$	$(a-b)(a+b) = a^2 - b^2$
$(a-b)^2 = a^2 - 2ab + b^2$

Squaring a binomial results in a trinomial.
Multiplying conjugates results in a binomial (a difference of squares).
With binomial squares, the inner and outer FOIL terms are the same.
With conjugates, the inner and outer FOIL terms are opposites.

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 6: Polynomials

Back to Openstax Elementary Algebra 2E

Lesson 1
Lesson 1: Add and Subtract Polynomials
Learn Practice
Lesson 2
Lesson 2: Use Multiplication Properties of Exponents
Learn Practice
Lesson 3
Lesson 3: Multiply Polynomials
Learn Practice
Lesson 4Current
Lesson 4: Special Products
Learn Practice
Lesson 5
Lesson 5: Divide Monomials
Learn Practice
Lesson 6
Lesson 6: Divide Polynomials
Learn Practice
Lesson 7
Lesson 7: Integer Exponents and Scientific Notation
Learn Practice

Lesson 4: Special Products

New Concept

What’s next

Property

Examples

Property

Examples

Property

Chapter 6: Polynomials

Lesson 1: Add and Subtract Polynomials

Lesson 2: Use Multiplication Properties of Exponents

Lesson 3: Multiply Polynomials

Lesson 4: Special Products

Lesson 5: Divide Monomials

Lesson 6: Divide Polynomials

Lesson 7: Integer Exponents and Scientific Notation

Lesson overview

New Concept

What’s next

Property

Examples

Property

Examples

Property

Chapter 6: Polynomials

Lesson 1: Add and Subtract Polynomials

Lesson 2: Use Multiplication Properties of Exponents

Lesson 3: Multiply Polynomials

Lesson 4: Special Products

Lesson 5: Divide Monomials

Lesson 6: Divide Polynomials

Lesson 7: Integer Exponents and Scientific Notation

Lesson 1: Add and Subtract Polynomials

Lesson 2: Use Multiplication Properties of Exponents

Lesson 3: Multiply Polynomials

Lesson 4: Special Products

Lesson 5: Divide Monomials

Lesson 6: Divide Polynomials

Lesson 7: Integer Exponents and Scientific Notation