Learn on PengiOpenstax Intermediate Algebra 2EChapter 5: Polynomials and Polynomial Functions

Lesson 5.3: Multiply Polynomials

New Concept Learn to multiply any type of polynomial, from simple monomials to trinomials. We'll cover key methods like the Distributive Property and FOIL, introduce time saving special product patterns, and apply these skills to polynomial functions.

Section 1

📘 Multiply Polynomials

New Concept

Learn to multiply any type of polynomial, from simple monomials to trinomials. We'll cover key methods like the Distributive Property and FOIL, introduce time-saving special product patterns, and apply these skills to polynomial functions.

What’s next

Next, we'll break down each multiplication method with interactive examples. You'll then test your skills with a sequence of practice cards.

Section 2

Multiply a Polynomial by a Monomial

Property

Multiplying a polynomial by a monomial is really just applying the Distributive Property.

Examples

Multiply $5x(3x^2 - 2x + 4)$ . Distribute $5x$ to each term: $5x(3x^2) + 5x(-2x) + 5x(4) = 15x^3 - 10x^2 + 20x$ .

Multiply $-4a^2b(a^2 - 3ab + 2b^2)$ . Distribute to all three terms: $(-4a^2b)(a^2) + (-4a^2b)(-3ab) + (-4a^2b)(2b^2) = -4a^4b + 12a^3b^2 - 8a^2b^3$ .

Section 3

FOIL Method for Binomials

Property

Use the FOIL method to multiply two binomials.

Step 1. Multiply the First terms.
Step 2. Multiply the Outer terms.
Step 3. Multiply the Inner terms.
Step 4. Multiply the Last terms.
Step 5. Combine like terms, when possible.

Examples

Multiply $(x+3)(x+6)$ . First: $x \cdot x = x^2$ . Outer: $x \cdot 6 = 6x$ . Inner: $3 \cdot x = 3x$ . Last: $3 \cdot 6 = 18$ . Combine: $x^2 + 6x + 3x + 18 = x^2 + 9x + 18$ .

Section 4

Multiply a Polynomial by a Polynomial

Property

To multiply a trinomial by a binomial, use the:

Distributive Property
Vertical Method

Remember, FOIL will not work in this case, but we can use either the Distributive Property or the Vertical Method.

Examples

Using the Distributive Property: $(x+4)(x^2+2x-3) = x(x^2+2x-3) + 4(x^2+2x-3) = x^3+2x^2-3x + 4x^2+8x-12 = x^3+6x^2+5x-12$ .

Section 5

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

To square a binomial, square the first term, square the last term, double their product.

Section 6

Product of Conjugates Pattern

Property

A conjugate pair is two binomials of the form $(a - b), (a + b)$ . 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.

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

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

Section 7

Multiply Polynomial Functions

Property

For functions $f(x)$ and $g(x)$ ,

(f \cdot g)(x) = f(x) \cdot g(x)

Examples

For $f(x)=x+4$ and $g(x)=x-6$ , find $(f \cdot g)(x)$ . We multiply: $(f \cdot g)(x) = (x+4)(x-6) = x^2-6x+4x-24 = x^2-2x-24$ .

Book overview

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

Continue this chapter

Chapter 5: Polynomials and Polynomial Functions

Back to Openstax Intermediate Algebra 2E

Lesson 1
Lesson 5.1: Add and Subtract Polynomials
Learn Practice
Lesson 2
Lesson 5.2: Properties of Exponents and Scientific Notation
Learn Practice
Lesson 3Current
Lesson 5.3: Multiply Polynomials
Learn Practice
Lesson 4
Lesson 5.4: Dividing Polynomials
Learn Practice

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

📘 Multiply Polynomials

New Concept

What’s next

Next, we'll break down each multiplication method with interactive examples. You'll then test your skills with a sequence of practice cards.

Section 2

Multiply a Polynomial by a Monomial

Property

Multiplying a polynomial by a monomial is really just applying the Distributive Property.

Examples

Multiply $5x(3x^2 - 2x + 4)$ . Distribute $5x$ to each term: $5x(3x^2) + 5x(-2x) + 5x(4) = 15x^3 - 10x^2 + 20x$ .

Multiply $-4a^2b(a^2 - 3ab + 2b^2)$ . Distribute to all three terms: $(-4a^2b)(a^2) + (-4a^2b)(-3ab) + (-4a^2b)(2b^2) = -4a^4b + 12a^3b^2 - 8a^2b^3$ .

Section 3

FOIL Method for Binomials

Property

Use the FOIL method to multiply two binomials.

Step 1. Multiply the First terms.
Step 2. Multiply the Outer terms.
Step 3. Multiply the Inner terms.
Step 4. Multiply the Last terms.
Step 5. Combine like terms, when possible.

Examples

Multiply $(x+3)(x+6)$ . First: $x \cdot x = x^2$ . Outer: $x \cdot 6 = 6x$ . Inner: $3 \cdot x = 3x$ . Last: $3 \cdot 6 = 18$ . Combine: $x^2 + 6x + 3x + 18 = x^2 + 9x + 18$ .

Section 4

Multiply a Polynomial by a Polynomial

Property

To multiply a trinomial by a binomial, use the:

Distributive Property
Vertical Method

Remember, FOIL will not work in this case, but we can use either the Distributive Property or the Vertical Method.

Examples

Using the Distributive Property: $(x+4)(x^2+2x-3) = x(x^2+2x-3) + 4(x^2+2x-3) = x^3+2x^2-3x + 4x^2+8x-12 = x^3+6x^2+5x-12$ .

Section 5

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

To square a binomial, square the first term, square the last term, double their product.

Section 6

Product of Conjugates Pattern

Property

A conjugate pair is two binomials of the form $(a - b), (a + b)$ . 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.

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

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

Section 7

Multiply Polynomial Functions

Property

For functions $f(x)$ and $g(x)$ ,

(f \cdot g)(x) = f(x) \cdot g(x)

Examples

For $f(x)=x+4$ and $g(x)=x-6$ , find $(f \cdot g)(x)$ . We multiply: $(f \cdot g)(x) = (x+4)(x-6) = x^2-6x+4x-24 = x^2-2x-24$ .

Book overview

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

Continue this chapter

Chapter 5: Polynomials and Polynomial Functions

Back to Openstax Intermediate Algebra 2E

Lesson 1
Lesson 5.1: Add and Subtract Polynomials
Learn Practice
Lesson 2
Lesson 5.2: Properties of Exponents and Scientific Notation
Learn Practice
Lesson 3Current
Lesson 5.3: Multiply Polynomials
Learn Practice
Lesson 4
Lesson 5.4: Dividing Polynomials
Learn Practice

Overview

Lesson overview

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

Overview

Section 1

📘 Multiply Polynomials

New Concept

What’s next

Next, we'll break down each multiplication method with interactive examples. You'll then test your skills with a sequence of practice cards.

Section 2

Multiply a Polynomial by a Monomial

Property

Multiplying a polynomial by a monomial is really just applying the Distributive Property.

Examples

Multiply $5x(3x^2 - 2x + 4)$ . Distribute $5x$ to each term: $5x(3x^2) + 5x(-2x) + 5x(4) = 15x^3 - 10x^2 + 20x$ .

Multiply $-4a^2b(a^2 - 3ab + 2b^2)$ . Distribute to all three terms: $(-4a^2b)(a^2) + (-4a^2b)(-3ab) + (-4a^2b)(2b^2) = -4a^4b + 12a^3b^2 - 8a^2b^3$ .

Section 3

FOIL Method for Binomials

Property

Use the FOIL method to multiply two binomials.

Step 1. Multiply the First terms.
Step 2. Multiply the Outer terms.
Step 3. Multiply the Inner terms.
Step 4. Multiply the Last terms.
Step 5. Combine like terms, when possible.

Examples

Multiply $(x+3)(x+6)$ . First: $x \cdot x = x^2$ . Outer: $x \cdot 6 = 6x$ . Inner: $3 \cdot x = 3x$ . Last: $3 \cdot 6 = 18$ . Combine: $x^2 + 6x + 3x + 18 = x^2 + 9x + 18$ .

Section 4

Multiply a Polynomial by a Polynomial

Property

To multiply a trinomial by a binomial, use the:

Distributive Property
Vertical Method

Remember, FOIL will not work in this case, but we can use either the Distributive Property or the Vertical Method.

Examples

Using the Distributive Property: $(x+4)(x^2+2x-3) = x(x^2+2x-3) + 4(x^2+2x-3) = x^3+2x^2-3x + 4x^2+8x-12 = x^3+6x^2+5x-12$ .

Section 5

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

To square a binomial, square the first term, square the last term, double their product.

Section 6

Product of Conjugates Pattern

Property

A conjugate pair is two binomials of the form $(a - b), (a + b)$ . 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.

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

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

Section 7

Multiply Polynomial Functions

Property

For functions $f(x)$ and $g(x)$ ,

(f \cdot g)(x) = f(x) \cdot g(x)

Examples

For $f(x)=x+4$ and $g(x)=x-6$ , find $(f \cdot g)(x)$ . We multiply: $(f \cdot g)(x) = (x+4)(x-6) = x^2-6x+4x-24 = x^2-2x-24$ .

Book overview

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

Continue this chapter

Chapter 5: Polynomials and Polynomial Functions

Back to Openstax Intermediate Algebra 2E

Lesson 1
Lesson 5.1: Add and Subtract Polynomials
Learn Practice
Lesson 2
Lesson 5.2: Properties of Exponents and Scientific Notation
Learn Practice
Lesson 3Current
Lesson 5.3: Multiply Polynomials
Learn Practice
Lesson 4
Lesson 5.4: Dividing Polynomials
Learn Practice

Lesson 5.3: Multiply Polynomials

New Concept

What’s next

Property

Examples

Property

Examples

Property

Examples

Property

Property

Property

Examples

Chapter 5: Polynomials and Polynomial Functions

Lesson 5.1: Add and Subtract Polynomials

Lesson 5.2: Properties of Exponents and Scientific Notation

Lesson 5.3: Multiply Polynomials

Lesson 5.4: Dividing Polynomials

Lesson overview

New Concept

What’s next

Property

Examples

Property

Examples

Property

Examples

Property

Property

Property

Examples

Chapter 5: Polynomials and Polynomial Functions

Lesson 5.1: Add and Subtract Polynomials

Lesson 5.2: Properties of Exponents and Scientific Notation

Lesson 5.3: Multiply Polynomials

Lesson 5.4: Dividing Polynomials

Lesson 5.1: Add and Subtract Polynomials

Lesson 5.2: Properties of Exponents and Scientific Notation

Lesson 5.3: Multiply Polynomials

Lesson 5.4: Dividing Polynomials