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Teacher Guide: Sum and Difference of Cubes

Learn to factor expressions in the form a³ + b³ and a³ - b³ using special formulas.

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10 questions on Factoring. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Recognize expressions that are sums or differences of perfect cubes
  • Apply the sum of cubes formula to factor a3+b3a^3 + b^3
  • Apply the difference of cubes formula to factor a3−b3a^3 - b^3
  • Use the SOAP method to remember the sign pattern
  • Factor expressions that require extracting a GCF before applying cube formulas
Prerequisites
  • • Understanding of exponents and perfect cubes
  • • Factoring out the greatest common factor
  • • Difference of squares factoring
  • • Expanding binomials and trinomials
Discussion Starters
  • 1. Why do you think the trinomial factor cannot be factored further?
  • 2. How does the SOAP method help you remember the formula? Can you think of another memory device?
  • 3. What would happen if we tried to apply these formulas to a3+b3+c3a^3 + b^3 + c^3?
  • 4. How are difference of squares and difference of cubes similar? How are they different?
Common Misconceptions

Trying to use (a+b)3=a3+b3(a + b)^3 = a^3 + b^3

Remediation: Show that (a+b)3=a3+3a2b+3ab2+b3≠a3+b3(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \neq a^3 + b^3. Have students expand to see the difference.

Thinking the trinomial factor can always be factored

Remediation: Use the discriminant to show why a2−ab+b2a^2 - ab + b^2 and a2+ab+b2a^2 + ab + b^2 have no real roots and thus cannot be factored.

Differentiation Ideas

For Struggling Students:

  • • Start with numerical examples before variables (8+278 + 27, 64−164 - 1)
  • • Provide a reference card with the formulas and SOAP pattern
  • • Practice identifying perfect cubes before factoring

For On-Level Students:

  • • Factor expressions with single variables and coefficients
  • • Practice mixed problems requiring GCF extraction first
  • • Verify answers by expanding the factored form

For Advanced Students:

  • • Factor expressions with multiple variables
  • • Explore the connection to complex number factoring
  • • Investigate sum and difference of higher powers
Standards Alignment
  • HSA-SSE.A.2 (CCSS.MATH.CONTENT.HSA.SSE.A.2)

    Use the structure of an expression to identify ways to rewrite it

  • HSA-APR.B.2 (CCSS.MATH.CONTENT.HSA.APR.B.2)

    Identify zeros of polynomials when suitable factorizations are available

Lesson Resources
  • visualCube Volume Visualization

    Interactive 3D cubes showing the geometric meaning of sum and difference of cubes

  • activitySOAP Sign Practice

    Quick-fire drill on identifying the correct signs in factored form

  • worksheetMixed Cube Factoring

    Practice problems mixing sum and difference of cubes with GCF factoring

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

The sum of cubes and difference of cubes are special factoring patterns for expressions where two perfect cubes are added or subtracted.

Sum of Cubes Formula

a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2)

Difference of Cubes Formula

a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2)
Memory Trick: SOAP
  • Same sign (as the original)
  • Opposite sign
  • Always Positive
In (a±b)(a2∓ab+b2)(a \pm b)(a^2 \mp ab + b^2):
  • First factor: same sign as original
  • Middle term of trinomial: opposite sign
  • Last term of trinomial: always positive

Worked Examples

Factor x3+8x^3 + 8
1

Identify the perfect cubes

x3=(x)3x^3 = (x)^3 and 8=238 = 2^3 → a=xa = x, b=2b = 2
2

Apply the sum of cubes formula

a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2) → (x+2)(x2−2x+4)(x + 2)(x^2 - 2x + 4)
3

Verify by expanding (optional)

(x+2)(x2−2x+4)=x3−2x2+4x+2x2−4x+8(x + 2)(x^2 - 2x + 4) = x^3 - 2x^2 + 4x + 2x^2 - 4x + 8 → x3+8x^3 + 8 \checkmark

Common Mistakes

Confusing the sign pattern in the trinomial

Why it's wrong: Students often forget whether the middle term is positive or negative.

Correct: Use SOAP: Same, Opposite, Always Positive. The middle term has the OPPOSITE sign from the binomial factor.

Thinking a2−ab+b2a^2 - ab + b^2 can be factored further

Why it's wrong: This trinomial looks like it might factor, but it cannot be factored over the real numbers.

Correct: The trinomials a2−ab+b2a^2 - ab + b^2 and a2+ab+b2a^2 + ab + b^2 are both prime (cannot be factored further).

Forgetting to identify cubes correctly

Why it's wrong: Students may not recognize numbers like 2727, 6464, or 125125 as perfect cubes.

Correct: Memorize the first ten cubes: 1,8,27,64,125,216,343,512,729,10001, 8, 27, 64, 125, 216, 343, 512, 729, 1000.

Not factoring out the GCF first

Why it's wrong: Jumping straight to cube formulas without checking for common factors.

Correct: Always check for a GCF before applying special factoring formulas.

Why It Matters

Sum and difference of cubes appear throughout advanced mathematics:
  • Simplifying algebraic expressions: Many complex expressions can be factored using these formulas
  • Solving cubic equations: Recognizing cube patterns helps solve equations like x3=27x^3 = 27
  • Calculus: These patterns appear when simplifying limits and derivatives
  • Physics and engineering: Volume calculations often involve cubic expressions
Mastering these formulas gives you powerful tools for simplifying expressions that would otherwise be very difficult to factor.

Real World Applications

Volume Calculations

When calculating the difference between two cubic containers, sum and difference of cubes formulas simplify the algebra.

Example:

A large cube has side length x+2x + 2 and a small cube has side length xx. The difference in volumes is (x+2)3−x3(x+2)^3 - x^3.

1Try It Yourself

A sculptor removes a cube with 3 cm sides from a cube with 5 cm sides.

Factor the expression for the remaining volume: 53−335^3 - 3^3

Step 1: Write the mathematical expression

Use the difference of cubes formula with a=5a = 5 and b=3b = 3:

Engineering and Physics

Cubic relationships appear in formulas for power, energy, and fluid dynamics.

Example:

The kinetic energy of wind is proportional to the cube of wind speed. Comparing two wind speeds involves cube expressions.

2Try It Yourself

Wind power at speed vv is proportional to v3v^3. How much more power does a 10 m/s wind have compared to a 4 m/s wind?

Simplify 103−4310^3 - 4^3 by factoring

Step 1: Write the mathematical expression

Apply difference of cubes:

Key Takeaways

  • 1Sum of cubes: a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2)
  • 2Difference of cubes: a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2)
  • 3Use SOAP to remember signs: Same, Opposite, Always Positive
  • 4Perfect cubes to memorize: 1,8,27,64,125,216,343,512,729,10001, 8, 27, 64, 125, 216, 343, 512, 729, 1000
  • 5The trinomial factor (a2±ab+b2a^2 \pm ab + b^2) cannot be factored further over real numbers
  • 6Always factor out the GCF before applying cube formulas

Frequently Asked Questions

Why can't we factor the trinomial further?

The discriminant of a2−ab+b2a^2 - ab + b^2 is b2−4(1)(b2)=−3b2<0b^2 - 4(1)(b^2) = -3b^2 < 0. Since it's negative, there are no real roots, so it cannot be factored over the real numbers.

How do I know if a number is a perfect cube?

Check if you can find an integer that, when cubed, equals the number. For example, 64=4364 = 4^3 because 4×4×4=644 \times 4 \times 4 = 64. Memorize common cubes: 1,8,27,64,125,216,343,512,729,10001, 8, 27, 64, 125, 216, 343, 512, 729, 1000.

What's the difference between difference of squares and difference of cubes?

Difference of squares: a2−b2=(a−b)(a+b)a^2 - b^2 = (a-b)(a+b) gives two binomials. Difference of cubes: a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a-b)(a^2+ab+b^2) gives a binomial and a trinomial.

Glossary

Perfect cube
A number that can be written as n3n^3 where nn is an integer (e.g., 8=238 = 2^3, 27=3327 = 3^3)
Sum of cubes
An expression in the form a3+b3a^3 + b^3, which factors as (a+b)(a2−ab+b2)(a + b)(a^2 - ab + b^2)
Difference of cubes
An expression in the form a3−b3a^3 - b^3, which factors as (a−b)(a2+ab+b2)(a - b)(a^2 + ab + b^2)
SOAP
Memory device: Same, Opposite, Always Positive - describes the signs in cube factoring formulas

Formula Card

Sum of Cubes

a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2)

Factor the sum of two cubes into a binomial and trinomial

Difference of Cubes

a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2)

Factor the difference of two cubes into a binomial and trinomial

SOAP Pattern

(a±b)(a2∓ab+b2)(a \pm b)(a^2 \mp ab + b^2)

Same, Opposite, Always Positive - sign pattern mnemonic

More in This Topic

Introduction to Factoring

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Factoring Out the GCF

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Factoring by Grouping

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Factoring Trinomials (a=1)

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