By Mark Koltko‑Rivera
Updated Aug 30, 2022

In algebra, a binomial is any expression with just two terms, like x + 5. When one or both terms are raised to the third power—such as x³ + 5 or y³ + 27—the expression becomes a cubic binomial. Simplifying these expressions is a common task in algebra, and it can be approached in three primary ways:

• 1. Cubing an entire binomial: (a + b)³ or (a – b)³
• 2. Cubing each term separately: a³ + b³ or a³ – b³
• 3. Other binomials where at least one term has degree three.

Below is a practical, formula‑driven walkthrough that ensures you handle each scenario with confidence.

Step 1: Identify the Type of Cubic Binomial

Determine which of the five basic categories you’re dealing with:

1. Cubing a binomial sum: (a + b)³
2. Cubing a binomial difference: (a – b)³
3. Sum of cubes: a³ + b³
4. Difference of cubes: a³ – b³
5. Any other binomial with a highest‑degree term of three.

Step 2: Use the Cubic Formula for a Sum

When expanding a sum, apply the binomial theorem: \[(a + b)³ = a³ + 3a²b + 3ab² + b³\]

Step 3: Use the Cubic Formula for a Difference

For a difference, the expansion is: \[(a – b)³ = a³ – 3a²b + 3ab² – b³\]

Step 4: Factor the Sum of Cubes

The sum of two cubes factors neatly: \[a³ + b³ = (a + b)(a² – ab + b²)\]

Step 5: Factor the Difference of Cubes

Similarly, the difference of cubes factors as: \[a³ – b³ = (a – b)(a² + ab + b²)\]

Step 6: Handle Other Cubic Binomials

Most binomials that don’t fit the above categories cannot be simplified further. The sole exception is when both terms share a variable, allowing you to factor out the lowest power. For example:

• x³ + x = x(x² + 1)
• x³ – x² = x²(x – 1)

These factorizations reduce the expression to a product of simpler terms, making further manipulation easier.

By following these steps, you’ll consistently arrive at the simplest form of any cubic binomial.