Learn on PengiOpenstax Elementary Algebra 2EChapter 7: Factoring

Lesson 4: Factor Special Products

In this lesson from OpenStax Elementary Algebra 2E, students learn to factor special products including perfect square trinomials, differences of squares, and sums and differences of cubes by recognizing patterns such as a² + 2ab + b² = (a+b)² and a² − b² = (a+b)(a−b). Students practice identifying when a polynomial fits one of these special forms and applying the corresponding pattern to factor it efficiently. The lesson also guides students in choosing the most appropriate factoring method to factor a polynomial completely.

Section 1

📘 Factor Special Products

New Concept

This lesson introduces shortcuts for factoring polynomials. By learning to recognize special patterns like perfect square trinomials, differences of squares, and sums/differences of cubes, you can factor expressions more quickly and efficiently.

What’s next

Now, let's break down each pattern. You'll work through interactive examples and practice cards to master factoring these special products.

Section 2

Factor Perfect Square Trinomials

Property

If $a$ and $b$ are real numbers, the perfect square trinomials pattern is as follows:

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

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

To use this pattern, first verify that the trinomial fits. Check if the first term is a perfect square (

a^2

) and the last term is a perfect square (

b^2

). Then, check if the middle term is twice their product (

2ab

). If it matches, write the square of the binomial

(a+b)^2

(a-b)^2

Examples

To factor $25x^2 + 30x + 9$ , recognize it as $(5x)^2 + 2(5x)(3) + 3^2$ . This fits the pattern $a^2+2ab+b^2$ , so the factored form is $(5x+3)^2$ .

To factor $49y^2 - 42y + 9$ , identify it as $(7y)^2 - 2(7y)(3) + 3^2$ . This matches the pattern $a^2-2ab+b^2$ , so the factored form is $(7y-3)^2$ .

Section 3

Factor Differences of Squares

Property

If $a$ and $b$ are real numbers, a difference of squares factors to a product of conjugates using the following pattern:

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

To use this pattern, ensure you have a binomial where two perfect squares are being subtracted. Write each term as a square,

(a)^2 - (b)^2

, then write the product of the conjugates,

(a-b)(a+b)

. Note that a sum of squares,

a^2+b^2

, is prime and cannot be factored.

Examples

To factor $9x^2 - 25$ , rewrite the expression as a difference of squares, $(3x)^2 - 5^2$ . This factors into the product of conjugates $(3x-5)(3x+5)$ .

To factor $16a^2 - 81b^2$ , identify the terms as $(4a)^2$ and $(9b)^2$ . The factored form is the product of their conjugates, $(4a-9b)(4a+9b)$ .

Section 4

Factor Sums and Differences of Cubes

Property

The patterns for factoring the sum and difference of cubes are:

a^3 + b^3 = (a+b)(a^2 - ab + b^2)

a^3 - b^3 = (a-b)(a^2 + ab + b^2)

To factor, confirm the binomial is a sum or difference of perfect cubes. Write the terms as cubes (

a^3

and

b^3

), then apply the corresponding pattern. The sign of the binomial factor matches the original binomial, while the middle sign of the trinomial factor is the opposite. The resulting trinomial factor is prime.

Examples

To factor $y^3 + 27$ , recognize it as a sum of cubes, $y^3 + 3^3$ . Applying the sum of cubes pattern gives $(y+3)(y^2 - 3y + 9)$ .

To factor $8x^3 - 1$ , see it as a difference of cubes, $(2x)^3 - 1^3$ . The pattern gives $(2x-1)((2x)^2 + (2x)(1) + 1^2)$ , which simplifies to $(2x-1)(4x^2 + 2x + 1)$ .

Section 5

Complete Factoring Strategy

Property

To factor a polynomial completely, follow a structured approach.

GCF First: Always factor out the Greatest Common Factor (GCF).
Identify Terms:
- Binomial (2 terms): Check for Difference of Squares ( $a^2-b^2$ ), Sum of Cubes ( $a^3+b^3$ ), or Difference of Cubes ( $a^3-b^3$ ).
- Trinomial (3 terms): Check for a Perfect Square Trinomial ( $a^2 \pm 2ab + b^2$ ).
Factor Further: Always check if any of the resulting factors can themselves be factored.

Examples

To factor $3x^3 - 12x$ , first factor out the GCF of $3x$ , which gives $3x(x^2 - 4)$ . The binomial is a difference of squares, so the final result is $3x(x-2)(x+2)$ .

To factor $2y^3 - 20y^2 + 50y$ , first factor out the GCF of $2y$ to get $2y(y^2 - 10y + 25)$ . The trinomial is a perfect square, $(y-5)^2$ . The result is $2y(y-5)^2$ .

Book overview

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

Continue this chapter

Chapter 7: Factoring

Back to Openstax Elementary Algebra 2E

Lesson 1
Lesson 1: Greatest Common Factor and Factor by Grouping
Learn Practice
Lesson 2
Lesson 2: Factor Trinomials of the Form x2+bx+c
Learn Practice
Lesson 3
Lesson 3: Factor Trinomials of the Form ax2+bx+c
Learn Practice
Lesson 4Current
Lesson 4: Factor Special Products
Learn Practice
Lesson 5
Lesson 5: General Strategy for Factoring Polynomials
Learn Practice
Lesson 6
Lesson 6: Quadratic Equations
Learn Practice

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

📘 Factor Special Products

New Concept

What’s next

Now, let's break down each pattern. You'll work through interactive examples and practice cards to master factoring these special products.

Section 2

Factor Perfect Square Trinomials

Property

If $a$ and $b$ are real numbers, the perfect square trinomials pattern is as follows:

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

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

To use this pattern, first verify that the trinomial fits. Check if the first term is a perfect square (

a^2

) and the last term is a perfect square (

b^2

). Then, check if the middle term is twice their product (

2ab

). If it matches, write the square of the binomial

(a+b)^2

(a-b)^2

Examples

To factor $25x^2 + 30x + 9$ , recognize it as $(5x)^2 + 2(5x)(3) + 3^2$ . This fits the pattern $a^2+2ab+b^2$ , so the factored form is $(5x+3)^2$ .

To factor $49y^2 - 42y + 9$ , identify it as $(7y)^2 - 2(7y)(3) + 3^2$ . This matches the pattern $a^2-2ab+b^2$ , so the factored form is $(7y-3)^2$ .

Section 3

Factor Differences of Squares

Property

If $a$ and $b$ are real numbers, a difference of squares factors to a product of conjugates using the following pattern:

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

To use this pattern, ensure you have a binomial where two perfect squares are being subtracted. Write each term as a square,

(a)^2 - (b)^2

, then write the product of the conjugates,

(a-b)(a+b)

. Note that a sum of squares,

a^2+b^2

, is prime and cannot be factored.

Examples

To factor $9x^2 - 25$ , rewrite the expression as a difference of squares, $(3x)^2 - 5^2$ . This factors into the product of conjugates $(3x-5)(3x+5)$ .

To factor $16a^2 - 81b^2$ , identify the terms as $(4a)^2$ and $(9b)^2$ . The factored form is the product of their conjugates, $(4a-9b)(4a+9b)$ .

Section 4

Factor Sums and Differences of Cubes

Property

The patterns for factoring the sum and difference of cubes are:

a^3 + b^3 = (a+b)(a^2 - ab + b^2)

a^3 - b^3 = (a-b)(a^2 + ab + b^2)

To factor, confirm the binomial is a sum or difference of perfect cubes. Write the terms as cubes (

a^3

and

b^3

Examples

To factor $y^3 + 27$ , recognize it as a sum of cubes, $y^3 + 3^3$ . Applying the sum of cubes pattern gives $(y+3)(y^2 - 3y + 9)$ .

To factor $8x^3 - 1$ , see it as a difference of cubes, $(2x)^3 - 1^3$ . The pattern gives $(2x-1)((2x)^2 + (2x)(1) + 1^2)$ , which simplifies to $(2x-1)(4x^2 + 2x + 1)$ .

Section 5

Complete Factoring Strategy

Property

To factor a polynomial completely, follow a structured approach.

GCF First: Always factor out the Greatest Common Factor (GCF).
Identify Terms:
- Binomial (2 terms): Check for Difference of Squares ( $a^2-b^2$ ), Sum of Cubes ( $a^3+b^3$ ), or Difference of Cubes ( $a^3-b^3$ ).
- Trinomial (3 terms): Check for a Perfect Square Trinomial ( $a^2 \pm 2ab + b^2$ ).
Factor Further: Always check if any of the resulting factors can themselves be factored.

Examples

To factor $3x^3 - 12x$ , first factor out the GCF of $3x$ , which gives $3x(x^2 - 4)$ . The binomial is a difference of squares, so the final result is $3x(x-2)(x+2)$ .

To factor $2y^3 - 20y^2 + 50y$ , first factor out the GCF of $2y$ to get $2y(y^2 - 10y + 25)$ . The trinomial is a perfect square, $(y-5)^2$ . The result is $2y(y-5)^2$ .

Book overview

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

Continue this chapter

Chapter 7: Factoring

Back to Openstax Elementary Algebra 2E

Lesson 1
Lesson 1: Greatest Common Factor and Factor by Grouping
Learn Practice
Lesson 2
Lesson 2: Factor Trinomials of the Form x2+bx+c
Learn Practice
Lesson 3
Lesson 3: Factor Trinomials of the Form ax2+bx+c
Learn Practice
Lesson 4Current
Lesson 4: Factor Special Products
Learn Practice
Lesson 5
Lesson 5: General Strategy for Factoring Polynomials
Learn Practice
Lesson 6
Lesson 6: Quadratic Equations
Learn Practice

Overview

Lesson overview

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

Overview

Section 1

📘 Factor Special Products

New Concept

What’s next

Now, let's break down each pattern. You'll work through interactive examples and practice cards to master factoring these special products.

Section 2

Factor Perfect Square Trinomials

Property

If $a$ and $b$ are real numbers, the perfect square trinomials pattern is as follows:

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

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

To use this pattern, first verify that the trinomial fits. Check if the first term is a perfect square (

a^2

) and the last term is a perfect square (

b^2

). Then, check if the middle term is twice their product (

2ab

). If it matches, write the square of the binomial

(a+b)^2

(a-b)^2

Examples

To factor $25x^2 + 30x + 9$ , recognize it as $(5x)^2 + 2(5x)(3) + 3^2$ . This fits the pattern $a^2+2ab+b^2$ , so the factored form is $(5x+3)^2$ .

To factor $49y^2 - 42y + 9$ , identify it as $(7y)^2 - 2(7y)(3) + 3^2$ . This matches the pattern $a^2-2ab+b^2$ , so the factored form is $(7y-3)^2$ .

Section 3

Factor Differences of Squares

Property

If $a$ and $b$ are real numbers, a difference of squares factors to a product of conjugates using the following pattern:

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

To use this pattern, ensure you have a binomial where two perfect squares are being subtracted. Write each term as a square,

(a)^2 - (b)^2

, then write the product of the conjugates,

(a-b)(a+b)

. Note that a sum of squares,

a^2+b^2

, is prime and cannot be factored.

Examples

To factor $9x^2 - 25$ , rewrite the expression as a difference of squares, $(3x)^2 - 5^2$ . This factors into the product of conjugates $(3x-5)(3x+5)$ .

To factor $16a^2 - 81b^2$ , identify the terms as $(4a)^2$ and $(9b)^2$ . The factored form is the product of their conjugates, $(4a-9b)(4a+9b)$ .

Section 4

Factor Sums and Differences of Cubes

Property

The patterns for factoring the sum and difference of cubes are:

a^3 + b^3 = (a+b)(a^2 - ab + b^2)

a^3 - b^3 = (a-b)(a^2 + ab + b^2)

To factor, confirm the binomial is a sum or difference of perfect cubes. Write the terms as cubes (

a^3

and

b^3

Examples

To factor $y^3 + 27$ , recognize it as a sum of cubes, $y^3 + 3^3$ . Applying the sum of cubes pattern gives $(y+3)(y^2 - 3y + 9)$ .

To factor $8x^3 - 1$ , see it as a difference of cubes, $(2x)^3 - 1^3$ . The pattern gives $(2x-1)((2x)^2 + (2x)(1) + 1^2)$ , which simplifies to $(2x-1)(4x^2 + 2x + 1)$ .

Section 5

Complete Factoring Strategy

Property

To factor a polynomial completely, follow a structured approach.

GCF First: Always factor out the Greatest Common Factor (GCF).
Identify Terms:
- Binomial (2 terms): Check for Difference of Squares ( $a^2-b^2$ ), Sum of Cubes ( $a^3+b^3$ ), or Difference of Cubes ( $a^3-b^3$ ).
- Trinomial (3 terms): Check for a Perfect Square Trinomial ( $a^2 \pm 2ab + b^2$ ).
Factor Further: Always check if any of the resulting factors can themselves be factored.

Examples

To factor $3x^3 - 12x$ , first factor out the GCF of $3x$ , which gives $3x(x^2 - 4)$ . The binomial is a difference of squares, so the final result is $3x(x-2)(x+2)$ .

To factor $2y^3 - 20y^2 + 50y$ , first factor out the GCF of $2y$ to get $2y(y^2 - 10y + 25)$ . The trinomial is a perfect square, $(y-5)^2$ . The result is $2y(y-5)^2$ .

Book overview

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

Continue this chapter

Chapter 7: Factoring

Back to Openstax Elementary Algebra 2E

Lesson 1
Lesson 1: Greatest Common Factor and Factor by Grouping
Learn Practice
Lesson 2
Lesson 2: Factor Trinomials of the Form x2+bx+c
Learn Practice
Lesson 3
Lesson 3: Factor Trinomials of the Form ax2+bx+c
Learn Practice
Lesson 4Current
Lesson 4: Factor Special Products
Learn Practice
Lesson 5
Lesson 5: General Strategy for Factoring Polynomials
Learn Practice
Lesson 6
Lesson 6: Quadratic Equations
Learn Practice

Lesson 4: Factor Special Products

New Concept

What’s next

Property

Examples

Property

Examples

Property

Examples

Property

Examples

Chapter 7: Factoring

Lesson 1: Greatest Common Factor and Factor by Grouping

Lesson 2: Factor Trinomials of the Form x2+bx+c

Lesson 3: Factor Trinomials of the Form ax2+bx+c

Lesson 4: Factor Special Products

Lesson 5: General Strategy for Factoring Polynomials

Lesson 6: Quadratic Equations

Lesson overview

New Concept

What’s next

Property

Examples

Property

Examples

Property

Examples

Property

Examples

Chapter 7: Factoring

Lesson 1: Greatest Common Factor and Factor by Grouping

Lesson 2: Factor Trinomials of the Form x2+bx+c

Lesson 3: Factor Trinomials of the Form ax2+bx+c

Lesson 4: Factor Special Products

Lesson 5: General Strategy for Factoring Polynomials

Lesson 6: Quadratic Equations

Lesson 1: Greatest Common Factor and Factor by Grouping

Lesson 2: Factor Trinomials of the Form x2+bx+c

Lesson 3: Factor Trinomials of the Form ax2+bx+c

Lesson 4: Factor Special Products

Lesson 5: General Strategy for Factoring Polynomials

Lesson 6: Quadratic Equations