# Cube of a Binomial - Formula & Worked Examples

TL;DR

The cube of a binomial expands as (a+b)³ = a³ + 3a²b + 3ab² + b³, with the middle coefficients 3 and 3 coming from how the three brackets multiply together. This article derives the formula, shows it as a physical cube split into eight blocks, covers the (a−b)³ version, works six examples, and names the term students drop most.

## What Is The Cube of a Binomial?

**The cube of a binomial is the result of raising a two-term expression to the third power, and it expands to four terms:** (a+b)³ = a³ + 3a²b + 3ab² + b³. The expansion is not a³ + b³; the two middle terms are always there, and dropping them is the single most common error.

### Variable Glossary:

| Symbol | Meaning |
| --- | --- |
| a | the first term of the binomial |
| b | the second term of the binomial |
| a³, b³ | the cubes of each term |
| 3a²b, 3ab² | the two cross-terms, where the coefficient 3 counts the ways each combination appears |

## How Is The Cube of a Binomial Formula Derived?

You derive it by multiplying step by step, never skipping a line. Start by writing the cube as a square times one more factor:

(a+b)³ = (a+b)²(a+b)

First expand the square:

(a+b)² = a² + 2ab + b²

Now multiply that trinomial by (a+b), distributing each term:

1. a²(a+b) = a³ + a²b 
2. 2ab(a+b) = 2a²b + 2ab²
3. b²(a+b) = ab² + b³

Add the three lines and combine like terms:

(a+b)³ = a³ + 3a²b + 3ab² + b³

## Examples of Cube of a Binomial

### Example 1

**Expand (x+2)³.**  
Match to (a+b)³ with a=x and b=2:

x³ + 3(x²)(2) + 3(x)(2²) + 2³ = x³ + 6x² + 12x + 8.  
**Final answer:** x³ + 6x² + 12x + 8.

### Example 2

**Expand (y−3)³.**  
Correctly use (a−b)³:

y³ - 3(y²)(3) + 3(y)(3²) - 3³ = y³ - 9y² + 27y - 27.  
**Final answer:** y³ - 9y² + 27y - 27.

### Example 3

**Expand (2a+1)³.**  
With first term 2a and second term 1:

(2a)³ + 3(2a)²(1) + 3(2a)(1)² + 1³ = 8a³ + 12a² + 6a + 1.  
**Final answer:** 8a³ + 12a² + 6a + 1.

### Example 4

**Expand (3x−2y)³.**  
With a=3x and b=2y:

(3x)³ - 3(3x)²(2y) + 3(3x)(2y)² - (2y)³ = 27x³ - 54x²y + 36xy² - 8y³.  
**Final answer:** 27x³ - 54x²y + 36xy² - 8y³.

### Example 5

**Use the cube of a binomial to evaluate 101³ mentally.** 
Write 101 as 100+1 and apply (a+b)³:

100³ + 3(100)²(1) + 3(100)(1²) + 1³ = 1,030,301.  
**Final answer:** 101³ = 1,030,301.

### Example 6

**Expand (x+12)³.**  
With a=x and b=12:

x³ + 3(x²)(12) + 3(x)(12²) + 12³ = x³ + 36x² + 54x + 1728.  
**Final answer:** x³ + 36x² + 54x + 1728.

## How Does This Relate To The Sum And Difference of Cubes?

The cube of a binomial expands a bracket _forward_. Its close cousin, the **sum and difference of cubes**, factors a two-term cubic expression:

- a³ + b³ = (a + b)(a² - ab + b²)
- a³ - b³ = (a - b)(a² + ab + b²)

Recognizing which form you are looking at — a cube _to expand_ or a sum/difference of cubes _to factor_ — is essential in any problem involving cubed terms.

## Mistakes Students Make

### Mistake 1: Dropping Middle Terms  
A binomial cube has four terms, not two; forgetting the cross-terms leads to incorrect results.

### Mistake 2: Forgetting the Coefficient 3  
Skipping the coefficient affects the accuracy of the terms.

### Mistake 3: Mishandling Signs  
For (a−b)³, remembering the signs alternate is crucial for correct expansion.

## Practice Questions on Cube of a Binomial

1. Expand (x+3)³.
2. Expand (a−4)³.
3. Expand (2y+1)³.
4. Expand (3x−2)³.
5. Use the formula to evaluate 993³.
6. Factor x³ + 27.

## Key Takeaways
- The cube of a binomial expands to four terms.
- The coefficients carry specific meanings in the expansion.
- Deriving the formula by hand helps avoid common mistakes.