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Difference of Cubes

Difference of cubes is a factoring pattern that breaks down an expression of the form a3−b3a^3 - b^3 into the product (a−b)(a2+ab+b2)(a - b)(a^2 + ab + b^2). It works whenever you're subtracting one perfect cube from another.

The difference of cubes is an algebraic identity stating that for any real numbers aa and bb, the expression a3−b3a^3 - b^3 factors as (a−b)(a2+ab+b2)(a - b)(a^2 + ab + b^2). This identity can be verified by expanding the right-hand side. The quadratic factor a2+ab+b2a^2 + ab + b^2 does not factor further over the real numbers, so the factored form is considered completely factored.

Key Formula

a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2)
Where:
  • aa = the cube root of the first term
  • bb = the cube root of the second term

Worked Example

Problem: Factor the expression 8x3−278x^3 - 27.
Step 1: Recognize each term as a perfect cube. Rewrite the expression so the cube structure is visible.
8x3=(2x)3and27=338x^3 = (2x)^3 \quad \text{and} \quad 27 = 3^3
Step 2: Identify aa and bb in the pattern a3−b3a^3 - b^3.
a=2x,b=3a = 2x, \quad b = 3
Step 3: Write the linear factor (a−b)(a - b).
(2x−3)(2x - 3)
Step 4: Build the trinomial factor a2+ab+b2a^2 + ab + b^2 by squaring aa, multiplying aa and bb, and squaring bb.
(2x)2+(2x)(3)+32=4x2+6x+9(2x)^2 + (2x)(3) + 3^2 = 4x^2 + 6x + 9
Step 5: Combine the two factors to write the fully factored form.
8x3−27=(2x−3)(4x2+6x+9)8x^3 - 27 = (2x - 3)(4x^2 + 6x + 9)
Answer: 8x3−27=(2x−3)(4x2+6x+9)8x^3 - 27 = (2x - 3)(4x^2 + 6x + 9)

Why It Matters

The difference of cubes formula appears frequently when solving polynomial equations in Algebra 2 and precalculus. Recognizing this pattern lets you factor expressions that would be extremely tedious to break down by trial and error. It also shows up in calculus when simplifying rational expressions and evaluating limits.

Common Mistakes

Mistake: Using a minus sign in the trinomial factor, writing (a2−ab+b2)(a^2 - ab + b^2) instead of (a2+ab+b2)(a^2 + ab + b^2).
Correction: The middle term of the trinomial factor has the opposite sign from the linear factor. Since the linear factor is (a−b)(a - b) with a minus, the trinomial uses a plus: a2+ab+b2a^2 + ab + b^2. A helpful mnemonic: the signs in the difference of cubes go "same, opposite, always positive" (SOAP).
Mistake: Trying to factor the trinomial a2+ab+b2a^2 + ab + b^2 further, as if it were a perfect square.
Correction: The expression a2+ab+b2a^2 + ab + b^2 is not the same as (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2. It has no real factors and should be left as is.

Related Terms

  • Factoring Rules — Difference of cubes is one standard factoring rule
  • Cube — The exponent structure underlying this pattern
  • Cube Root — Used to identify a and b in the formula