Learn on PengiAoPS: Introduction to Algebra (AMC 8 & 10)Chapter 11: Special Factorizations

Lesson 3: Sum and Difference of Cubes

In this Grade 4 AoPS Introduction to Algebra lesson from Chapter 11, students learn how to factor the sum and difference of cubes using the identities x³ − y³ = (x − y)(x² + xy + y²) and x³ + y³ = (x + y)(x² − xy + y²). The lesson guides students through deriving these factorizations by recognizing patterns in expanded products, then applying logical reasoning to construct the correct quadratic factors. Special attention is given to understanding why each identity works rather than simply memorizing the formulas, helping students avoid common errors when applying these special factorizations.

Section 1

Sum and Difference of Cubes Patterns

Property

Sum of Cubes: $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$
Difference of Cubes: $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$

Examples

Section 2

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 3

Common Errors in Cube Factorization

Property

Common cube factorization errors to avoid:

Confusing signs: $x^3 + y^3 = (x + y)(x^2 - xy + y^2)$ vs $x^3 - y^3 = (x - y)(x^2 + xy + y^2)$
Incorrect quadratic factors: $(x + y)(x^2 + xy + y^2)$ is wrong for sum of cubes
Missing the middle term $xy$ in the quadratic factor

Examples

Book overview

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Chapter 11: Special Factorizations

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Lesson 1
Lesson 1: Squares of Binomials
Learn Practice
Lesson 2
Lesson 2: Difference of Squares
Learn Practice
Lesson 3Current
Lesson 3: Sum and Difference of Cubes
Learn Practice
Lesson 4
Lesson 4: Rationalizing Denominators
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Lesson 5
Lesson 5: Simon's Favorite Factoring Trick
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Lesson overview

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Section 1

Sum and Difference of Cubes Patterns

Property

Sum of Cubes: $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$
Difference of Cubes: $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$

Examples

Section 2

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 3

Common Errors in Cube Factorization

Property

Common cube factorization errors to avoid:

Confusing signs: $x^3 + y^3 = (x + y)(x^2 - xy + y^2)$ vs $x^3 - y^3 = (x - y)(x^2 + xy + y^2)$
Incorrect quadratic factors: $(x + y)(x^2 + xy + y^2)$ is wrong for sum of cubes
Missing the middle term $xy$ in the quadratic factor

Examples

Book overview

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

Continue this chapter

Chapter 11: Special Factorizations

Back to AoPS: Introduction to Algebra (AMC 8 & 10)

Lesson 1
Lesson 1: Squares of Binomials
Learn Practice
Lesson 2
Lesson 2: Difference of Squares
Learn Practice
Lesson 3Current
Lesson 3: Sum and Difference of Cubes
Learn Practice
Lesson 4
Lesson 4: Rationalizing Denominators
Learn Practice
Lesson 5
Lesson 5: Simon's Favorite Factoring Trick
Learn Practice

Overview

Lesson overview

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

Overview

Section 1

Sum and Difference of Cubes Patterns

Property

Sum of Cubes: $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$
Difference of Cubes: $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$

Examples

Section 2

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 3

Common Errors in Cube Factorization

Property

Common cube factorization errors to avoid:

Confusing signs: $x^3 + y^3 = (x + y)(x^2 - xy + y^2)$ vs $x^3 - y^3 = (x - y)(x^2 + xy + y^2)$
Incorrect quadratic factors: $(x + y)(x^2 + xy + y^2)$ is wrong for sum of cubes
Missing the middle term $xy$ in the quadratic factor

Examples

Book overview

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

Continue this chapter

Chapter 11: Special Factorizations

Back to AoPS: Introduction to Algebra (AMC 8 & 10)

Lesson 1
Lesson 1: Squares of Binomials
Learn Practice
Lesson 2
Lesson 2: Difference of Squares
Learn Practice
Lesson 3Current
Lesson 3: Sum and Difference of Cubes
Learn Practice
Lesson 4
Lesson 4: Rationalizing Denominators
Learn Practice
Lesson 5
Lesson 5: Simon's Favorite Factoring Trick
Learn Practice

Lesson 3: Sum and Difference of Cubes

Property

Examples

Property

Examples

Property

Examples

Chapter 11: Special Factorizations

Lesson 1: Squares of Binomials

Lesson 2: Difference of Squares

Lesson 3: Sum and Difference of Cubes

Lesson 4: Rationalizing Denominators

Lesson 5: Simon's Favorite Factoring Trick

Lesson overview

Property

Examples

Property

Examples

Property

Examples

Chapter 11: Special Factorizations

Lesson 1: Squares of Binomials

Lesson 2: Difference of Squares

Lesson 3: Sum and Difference of Cubes

Lesson 4: Rationalizing Denominators

Lesson 5: Simon's Favorite Factoring Trick

Lesson 1: Squares of Binomials

Lesson 2: Difference of Squares

Lesson 3: Sum and Difference of Cubes

Lesson 4: Rationalizing Denominators

Lesson 5: Simon's Favorite Factoring Trick