Sum and Difference of Cubes

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

Advanced25 minLesson

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

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

Difference of Cubes Formula

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

Memory Trick: SOAP

Same sign (as the original)
Opposite sign
Always Positive

(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 $x^3 + 8$

Identify the perfect cubes

$x^3 = (x)^3$ and $8 = 2^3$ → $a = x$ , $b = 2$

Apply the sum of cubes formula

$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ → $(x + 2)(x^2 - 2x + 4)$

Verify by expanding (optional)

$(x + 2)(x^2 - 2x + 4) = x^3 - 2x^2 + 4x + 2x^2 - 4x + 8$ → $x^3 + 8$ \checkmark

Answer: $(x + 2)(x^2 - 2x + 4)$

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 $a^2 - ab + b^2$ 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 $a^2 - ab + b^2$ and $a^2 + ab + b^2$ are both prime (cannot be factored further).

Forgetting to identify cubes correctly

Why it's wrong: Students may not recognize numbers like $27$ , $64$ , or $125$ as perfect cubes.

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

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.

Interactive Visual

3D Shape Viewer

Select Shape

Rotate X (20°)

Rotate Y (30°)

Length (4)

Faces

Edges

Vertices

Volume

V = s³

64 units³

Surface Area

SA = 6s²

96 units²

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 $x^3 = 27$
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 $x + 2$ and a small cube has side length $x$ . The difference in volumes is $(x+2)^3 - x^3$ .

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: $5^3 - 3^3$

Step 1: Write the mathematical expression

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

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 $v$ is proportional to $v^3$ . How much more power does a 10 m/s wind have compared to a 4 m/s wind?

Simplify $10^3 - 4^3$ by factoring

Step 1: Write the mathematical expression

Apply difference of cubes:

Key Takeaways

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

Frequently Asked Questions

The discriminant of

a^2 - ab + b^2

b^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.

Glossary

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

Formula Card

Sum of Cubes

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

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 \pm b)(a^2 \mp ab + b^2)

Same, Opposite, Always Positive - sign pattern mnemonic