Learn on PengiYoshiwara Elementary AlgebraChapter 5: Exponents and Roots

Lesson 4: Products of Binomials

New Concept This lesson teaches you how to multiply binomials. You'll learn to distribute each term from the first binomial to every term in the second. This process can be visualized using an area model, resulting in an expanded polynomial.

Section 1

📘 Products of Binomials

New Concept

This lesson teaches you how to multiply binomials. You'll learn to distribute each term from the first binomial to every term in the second. This process can be visualized using an area model, resulting in an expanded polynomial.

What’s next

Soon, you'll use our interactive tools to see how the area model works. Then, you'll master the process with a series of practice cards.

Section 2

Products of variables

Property

The commutative and associative properties tell us that we can multiply the factors of a product in any order. To simplify a product like $(3b)(4b)$ , we can rearrange and group the factors:

(3b)(4b) = (3 \cdot 4) \cdot (b \cdot b) = 12b^2

Examples

To simplify $(6a)(-4a)$ , we multiply the coefficients and the variables separately: $(6)(-4) \cdot (a)(a) = -24a^2$ .

To simplify $(3x)^3$ , we multiply three copies of the expression: $(3x)(3x)(3x) = (3 \cdot 3 \cdot 3) \cdot (x \cdot x \cdot x) = 27x^3$ .

Section 3

Adding versus multiplying terms

Property

When we add like terms, we do not change the variable in the terms; we combine the coefficients. For example:

3a + 2a = 5a

When we multiply expressions, we multiply the coefficients and we multiply the variables:

3a(2a) = 3(2) \cdot a \cdot a = 6a^2

Examples

Adding: $8x + 3x = 11x$ . Multiplying: $(8x)(3x) = 24x^2$ .

Adding: $10y^2 + 5y^2 = 15y^2$ . Multiplying: $(10y^2)(5y^2) = 50y^4$ .

Section 4

Using the distributive law

Property

The distributive law states that for any numbers $a$ , $b$ , and $c$ :

a(b + c) = ab + ac

This means you can 'distribute' the multiplication to each term inside a set of parentheses.

Examples

To find the product $6(x+4)$ , we distribute the 6 to both terms inside: $6(x) + 6(4) = 6x + 24$ .

For $5a(3a-2)$ , we distribute the $5a$ : $5a(3a) - 5a(2) = 15a^2 - 10a$ .

Section 5

Types of algebraic expressions

Property

• An algebraic expression with only one term, such as $2x^4$ , is called a monomial.
• An expression with two terms, such as $x^2 - 16$ , is called a binomial.
• An expression with three terms is a trinomial.

Examples

A monomial has one term, like $8x^3$ or $-12$ .

A binomial has two terms, like $5x - 9$ or $a^2 + b^2$ .

Section 6

Multiplying binomials

Property

To multiply two binomials, multiply each term of the first binomial by each term of the second binomial. The acronym FOIL helps organize the four products:

First terms
Outer terms
Inner terms
Last terms

(x-4)(x+6) = x^2 + 6x - 4x - 24 = x^2 + 2x - 24

Examples

Using FOIL for $(x+2)(x+7)$ : $x^2$ (F) $+ 7x$ (O) $+ 2x$ (I) $+ 14$ (L), which simplifies to $x^2 + 9x + 14$ .

For $(3y-2)(y+5)$ : $3y^2$ (F) $+ 15y$ (O) $- 2y$ (I) $- 10$ (L), which simplifies to $3y^2 + 13y - 10$ .

Book overview

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Chapter 5: Exponents and Roots

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Lesson 1
Lesson 1: Exponents
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Lesson 2
Lesson 2: Square Roots and Cube Roots
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Lesson 3
Lesson 3: Using Formulas
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Lesson 4Current
Lesson 4: Products of Binomials
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Lesson 5
Lesson 5.5: Chapter Summary and Review
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Lesson overview

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Section 1

📘 Products of Binomials

New Concept

What’s next

Soon, you'll use our interactive tools to see how the area model works. Then, you'll master the process with a series of practice cards.

Section 2

Products of variables

Property

The commutative and associative properties tell us that we can multiply the factors of a product in any order. To simplify a product like $(3b)(4b)$ , we can rearrange and group the factors:

(3b)(4b) = (3 \cdot 4) \cdot (b \cdot b) = 12b^2

Examples

To simplify $(6a)(-4a)$ , we multiply the coefficients and the variables separately: $(6)(-4) \cdot (a)(a) = -24a^2$ .

To simplify $(3x)^3$ , we multiply three copies of the expression: $(3x)(3x)(3x) = (3 \cdot 3 \cdot 3) \cdot (x \cdot x \cdot x) = 27x^3$ .

Section 3

Adding versus multiplying terms

Property

When we add like terms, we do not change the variable in the terms; we combine the coefficients. For example:

3a + 2a = 5a

When we multiply expressions, we multiply the coefficients and we multiply the variables:

3a(2a) = 3(2) \cdot a \cdot a = 6a^2

Examples

Adding: $8x + 3x = 11x$ . Multiplying: $(8x)(3x) = 24x^2$ .

Adding: $10y^2 + 5y^2 = 15y^2$ . Multiplying: $(10y^2)(5y^2) = 50y^4$ .

Section 4

Using the distributive law

Property

The distributive law states that for any numbers $a$ , $b$ , and $c$ :

a(b + c) = ab + ac

This means you can 'distribute' the multiplication to each term inside a set of parentheses.

Examples

To find the product $6(x+4)$ , we distribute the 6 to both terms inside: $6(x) + 6(4) = 6x + 24$ .

For $5a(3a-2)$ , we distribute the $5a$ : $5a(3a) - 5a(2) = 15a^2 - 10a$ .

Section 5

Types of algebraic expressions

Property

Examples

A monomial has one term, like $8x^3$ or $-12$ .

A binomial has two terms, like $5x - 9$ or $a^2 + b^2$ .

Section 6

Multiplying binomials

Property

To multiply two binomials, multiply each term of the first binomial by each term of the second binomial. The acronym FOIL helps organize the four products:

First terms
Outer terms
Inner terms
Last terms

(x-4)(x+6) = x^2 + 6x - 4x - 24 = x^2 + 2x - 24

Examples

Using FOIL for $(x+2)(x+7)$ : $x^2$ (F) $+ 7x$ (O) $+ 2x$ (I) $+ 14$ (L), which simplifies to $x^2 + 9x + 14$ .

For $(3y-2)(y+5)$ : $3y^2$ (F) $+ 15y$ (O) $- 2y$ (I) $- 10$ (L), which simplifies to $3y^2 + 13y - 10$ .

Book overview

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

Continue this chapter

Chapter 5: Exponents and Roots

Back to Yoshiwara Elementary Algebra

Lesson 1
Lesson 1: Exponents
Learn Practice
Lesson 2
Lesson 2: Square Roots and Cube Roots
Learn Practice
Lesson 3
Lesson 3: Using Formulas
Learn Practice
Lesson 4Current
Lesson 4: Products of Binomials
Learn Practice
Lesson 5
Lesson 5.5: Chapter Summary and Review
Learn Practice

Overview

Lesson overview

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

Overview

Section 1

📘 Products of Binomials

New Concept

What’s next

Soon, you'll use our interactive tools to see how the area model works. Then, you'll master the process with a series of practice cards.

Section 2

Products of variables

Property

The commutative and associative properties tell us that we can multiply the factors of a product in any order. To simplify a product like $(3b)(4b)$ , we can rearrange and group the factors:

(3b)(4b) = (3 \cdot 4) \cdot (b \cdot b) = 12b^2

Examples

To simplify $(6a)(-4a)$ , we multiply the coefficients and the variables separately: $(6)(-4) \cdot (a)(a) = -24a^2$ .

To simplify $(3x)^3$ , we multiply three copies of the expression: $(3x)(3x)(3x) = (3 \cdot 3 \cdot 3) \cdot (x \cdot x \cdot x) = 27x^3$ .

Section 3

Adding versus multiplying terms

Property

When we add like terms, we do not change the variable in the terms; we combine the coefficients. For example:

3a + 2a = 5a

When we multiply expressions, we multiply the coefficients and we multiply the variables:

3a(2a) = 3(2) \cdot a \cdot a = 6a^2

Examples

Adding: $8x + 3x = 11x$ . Multiplying: $(8x)(3x) = 24x^2$ .

Adding: $10y^2 + 5y^2 = 15y^2$ . Multiplying: $(10y^2)(5y^2) = 50y^4$ .

Section 4

Using the distributive law

Property

The distributive law states that for any numbers $a$ , $b$ , and $c$ :

a(b + c) = ab + ac

This means you can 'distribute' the multiplication to each term inside a set of parentheses.

Examples

To find the product $6(x+4)$ , we distribute the 6 to both terms inside: $6(x) + 6(4) = 6x + 24$ .

For $5a(3a-2)$ , we distribute the $5a$ : $5a(3a) - 5a(2) = 15a^2 - 10a$ .

Section 5

Types of algebraic expressions

Property

Examples

A monomial has one term, like $8x^3$ or $-12$ .

A binomial has two terms, like $5x - 9$ or $a^2 + b^2$ .

Section 6

Multiplying binomials

Property

To multiply two binomials, multiply each term of the first binomial by each term of the second binomial. The acronym FOIL helps organize the four products:

First terms
Outer terms
Inner terms
Last terms

(x-4)(x+6) = x^2 + 6x - 4x - 24 = x^2 + 2x - 24

Examples

Using FOIL for $(x+2)(x+7)$ : $x^2$ (F) $+ 7x$ (O) $+ 2x$ (I) $+ 14$ (L), which simplifies to $x^2 + 9x + 14$ .

For $(3y-2)(y+5)$ : $3y^2$ (F) $+ 15y$ (O) $- 2y$ (I) $- 10$ (L), which simplifies to $3y^2 + 13y - 10$ .

Book overview

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

Continue this chapter

Chapter 5: Exponents and Roots

Back to Yoshiwara Elementary Algebra

Lesson 1
Lesson 1: Exponents
Learn Practice
Lesson 2
Lesson 2: Square Roots and Cube Roots
Learn Practice
Lesson 3
Lesson 3: Using Formulas
Learn Practice
Lesson 4Current
Lesson 4: Products of Binomials
Learn Practice
Lesson 5
Lesson 5.5: Chapter Summary and Review
Learn Practice

Lesson 4: Products of Binomials

New Concept

What’s next

Property

Examples

Property

Examples

Property

Examples

Property

Examples

Property

Examples

Chapter 5: Exponents and Roots

Lesson 1: Exponents

Lesson 2: Square Roots and Cube Roots

Lesson 3: Using Formulas

Lesson 4: Products of Binomials

Lesson 5.5: Chapter Summary and Review

Lesson overview

New Concept

What’s next

Property

Examples

Property

Examples

Property

Examples

Property

Examples

Property

Examples

Chapter 5: Exponents and Roots

Lesson 1: Exponents

Lesson 2: Square Roots and Cube Roots

Lesson 3: Using Formulas

Lesson 4: Products of Binomials

Lesson 5.5: Chapter Summary and Review

Lesson 1: Exponents

Lesson 2: Square Roots and Cube Roots

Lesson 3: Using Formulas

Lesson 4: Products of Binomials

Lesson 5.5: Chapter Summary and Review