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:

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.