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Sum of Cubes

Sum of cubes is a factoring formula that breaks the expression a3+b3a^3 + b^3 into the product (a+b)(a2−ab+b2)(a + b)(a^2 - ab + b^2). It gives you a way to factor any expression that can be written as one perfect cube added to another.

The sum of cubes identity states that for any real numbers aa and bb, the polynomial a3+b3a^3 + b^3 factors as (a+b)(a2−ab+b2)(a + b)(a^2 - ab + b^2). This factorization is irreducible over the reals, meaning the trinomial a2−ab+b2a^2 - ab + b^2 cannot be factored further using real numbers. The identity can be verified by expanding the right-hand side and confirming it equals the left-hand side.

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: Rewrite each term as a perfect cube. Recognize that 8x3=(2x)38x^3 = (2x)^3 and 27=3327 = 3^3.
(2x)3+33(2x)^3 + 3^3
Step 2: Identify aa and bb from the pattern. Here a=2xa = 2x and b=3b = 3.
Step 3: Write the binomial factor (a+b)(a + b).
(2x+3)(2x + 3)
Step 4: Build the trinomial factor (a2−ab+b2)(a^2 - ab + b^2) by substituting a=2xa = 2x and b=3b = 3.
(2x)2−(2x)(3)+32=4x2−6x+9(2x)^2 - (2x)(3) + 3^2 = 4x^2 - 6x + 9
Step 5: Combine both factors to write the final answer.
(2x+3)(4x2−6x+9)(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 sum of cubes formula appears frequently in Algebra 2 and precalculus when simplifying rational expressions or solving polynomial equations. Engineers and physicists use it when simplifying cubic relationships in formulas. Recognizing this pattern quickly saves significant time on tests and makes more complex factoring problems manageable.

Common Mistakes

Mistake: Using the wrong sign in the trinomial factor — writing a2+ab+b2a^2 + ab + b^2 instead of a2−ab+b2a^2 - ab + b^2.
Correction: Remember the mnemonic SOAP: Same sign, Opposite sign, Always Positive. The binomial gets the Same sign as the original (++), the middle term of the trinomial gets the Opposite sign (−-), and the last term is Always Positive (++).
Mistake: Confusing sum of cubes with difference of cubes and applying the wrong formula.
Correction: For a3+b3a^3 + b^3, the binomial factor is (a+b)(a + b). For a3−b3a^3 - b^3, the binomial factor is (a−b)(a - b). Double-check the operation between the two cubes before factoring.

Related Terms

  • Factoring Rules — Sum of cubes is one of the key factoring rules
  • Cube — Each term in the expression is a perfect cube
  • Cube Root — Used to identify aa and bb in the formula