Sum and Difference of Cubes

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

Advanced25 minLesson
Printable worksheet

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

Try it now

Which of the following is a perfect cube?

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.

Watch a video explanation

The same topic explained by another teacher, if a video helps you more.

Factoring Sums and Differences of Cubes | Step-by-Step Examples

by Understand The Math

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Interactive Visual

Composite solid

Volume

  • Cube: V = s³ = 27 units³
  • − Cube (hole): V = s³ = 8 units³

Total volume: 19 units³

The volume of a composite solid is the sum of the volumes of its parts. A hole is subtracted.

Surface Area

  • Cube: 54 − 4 (openings) = 50 units²
  • + Cube (inner wall): 24 − 4 (openings) = 20 units²

Total surface area: 70 units²

Surface area counts only the outside: where two parts touch, that face is hidden and is taken away from both parts. A hole takes away its openings and adds its inner wall.

Function graph

-4-3-2-10123-20-100102030xy(-2, 0)

Tap or hover over the graph to read off values.

x^3 + 8(x + 2)(x^2 - 2x + 4)
  • Zero (x-intercept): (-2, 0)
  • Zero (x-intercept): (-2, 0)

Interactive Sandbox

Expression Calculator

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Practice Problems

18 problems
Problem 1 of 18
Easy
Which of the following is a perfect cube?

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

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.
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.
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.
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

Today's challengeOne problem, three tries

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