Cube of Difference Identity with Algebraic Examples

Cube of Difference Identity expands the cube of a binomial in the form (a − b)³. The expansion contains four terms with coefficients 1, 3, 3, 1 and alternating signs. This page explains the formula, sign order, direct substitution, algebraic expansion and common cube of difference identity problems.

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What Is the Cube of Difference Identity?

The cube of difference identity is (a − b)³ = a³ − 3a²b + 3ab² − b³. It gives the expansion of the cube of the difference between two quantities.

The terms are formed from a³, a²b, ab² and b³. Their coefficients follow 1, 3, 3, 1, while the signs alternate as positive, negative, positive, negative. For example, (x − 2)³ = x³ − 6x² + 12x − 8.

Cube of Difference Identity Video Lessons

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Cube Identities: A Plus or Minus B

Learn the cube identities for (A+B)³ and (A−B)³, along with the factorisations of A³+B³ and A³−B³ for solving algebraic expressions.

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Cube of Difference Identity Formula & Tricks

Important Formulas

Cube of a Difference
(a − b)³ = a³ − 3a²b + 3ab² − b³

Use this identity when a binomial is raised to the third power and its terms are separated by a minus sign.

Equivalent Factor Form
a³ − 3a²b + 3ab² − b³ = (a − b)³

The expanded expression can be recognised and written as a perfect cube.

Relation with Sum and Difference of Cubes
(a − b)³ = a³ − b³ − 3ab(a − b)

This form is obtained by grouping the first and last cube terms in the standard expansion.

Quick Tricks

Use the 1-3-3-1 coefficient pattern

Write the powers of the first term from 3 to 0 and the powers of the second term from 0 to 3. Multiply the middle terms by 3, and use the sign pattern +, −, +, −.

Example: (x − 4)³ = x³ − 3(x²)(4) + 3(x)(4²) − 4³ = x³ − 12x² + 48x − 64.
Check the result using a simple substitution

For an algebraic expansion, assign small values to the variables and compare both sides. This helps detect sign or coefficient errors.

Example: For a = 5 and b = 2, the left side is 3³ = 27. The right side is 125 − 150 + 60 − 8 = 27.

Cube of Difference Identity Concepts

Direct Application of the Identity

Substitute the two quantities directly into (a − b)³ = a³ − 3a²b + 3ab² − b³.

Keep the first quantity as a and the subtracted quantity as b. Calculate the four terms separately before combining them. This avoids errors in the middle terms.

Example: (7 − 2)³ = 7³ − 3(7²)(2) + 3(7)(2²) − 2³ = 343 − 294 + 84 − 8 = 125.

Expansion of Algebraic Binomials

For an expression such as (x − k)³, replace a with x and b with k, then simplify each term.

The general expansion is x³ − 3kx² + 3k²x − k³. If the second term contains a variable, multiply the powers carefully.

Example: (2x − 3)³ = (2x)³ − 3(2x)²(3) + 3(2x)(3²) − 3³ = 8x³ − 36x² + 54x − 27.

Finding a Perfect Cube from Its Expansion

An expression has the form (a − b)³ when its terms match a³ − 3a²b + 3ab² − b³.

The first and last terms identify a and b through their cubes. The two middle terms must have coefficients and signs consistent with 3a²b and 3ab².

Example: x³ − 9x² + 27x − 27 = x³ − 3(x²)(3) + 3(x)(3²) − 3³ = (x − 3)³.

Using the Identity for Numerical Calculation

Express a number as the difference of two convenient numbers, then apply the identity instead of multiplying the full cube directly.

Choose a and b so that a − b equals the required number and the powers of a and b are easy to calculate.

Example: 49³ = (50 − 1)³ = 50³ − 3(50²)(1) + 3(50)(1²) − 1³ = 125000 − 7500 + 150 − 1 = 117649.

Cube of Difference Identity Data Table

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Term and Sign Pattern

The four terms in the cube of difference identity follow descending powers of a and ascending powers of b.

Term position Term Sign Coefficient
1 a³ + 1
2 3a²b − 3
3 3ab² + 3
4 b³ − 1

Note: The signs are +, −, +, − when the expression is (a − b)³.

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Practice Cube of Difference Identity Questions

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1The price of petrol is increased by 28%. A person wants to increase his expenditure by 22% only. By approximately what percent should he decrease his consumption?→ 2An article is sold for Rs. 288 after successive discounts of 20% and 25%. What is the marked price of the article?→ 321.6 ÷ 3.6 × 2 + 0.25 × 16 ÷ 4 - 6 is equal to∶→ 4What is the value of x so that the seven digit number 55350x2 is divisible by 72?→ 5Two articles are sold for Rs. 4,752 each. On one, the seller gains 32% and on the other he loses 28%. What is his overall gain or loss percentage, correct to one decimal place?→ 6The price of sugar is increased by 17%. A person wants to increase his expenditure by 5% only. By approximately what percent should he decrease his consumption?→ 7The value of 15.2 + 5.8 ÷ 2.9 × 2 - 3.5 × 2 ÷ 0.5 is equal to:→ 8What is the value of x so that the seven digit number 8439x53 is divisible by 99?→ 9Two articles are sold for Rs. 962 each. On one, the seller gains 30% and on the other he loses 26%. What is his overall gain or loss percentage, nearest to one decimal place?→ 10An article is sold for Rs. 288 after successive discounts of 25% and x%. If the marked price of the article is Rs. 480, what is the value of x?→

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Quick Revision Notes

Cube of Difference Identity Revision Points

Recall the formula, coefficient pattern and conditions for identifying a perfect cube.

  • (a − b)³ = a³ − 3a²b + 3ab² − b³.
  • The coefficients are 1, 3, 3, 1.
  • The signs alternate as +, −, +, −.
  • The powers of a decrease from 3 to 0, while the powers of b increase from 0 to 3.
  • For (x − k)³, the expansion is x³ − 3kx² + 3k²x − k³.
  • Do not confuse (a − b)³ with a³ − b³; the latter omits the two middle terms.

Related Topic Explainers

Substitution Method for Simultaneous Equations → Cross Multiplication Method for Linear Equations → Solving Quadratic Equations by Factorisation → Quadratic Formula → Discriminant of a Quadratic Equation → Nature of Roots of a Quadratic Equation → Sum and Product of Roots of Quadratic Equation → Quadratic Inequality →

Cube of Difference Identity FAQs

What is the cube of difference identity formula?

The formula is (a − b)³ = a³ − 3a²b + 3ab² − b³.

How do you expand (x − 5)³?

(x − 5)³ = x³ − 3(x²)(5) + 3(x)(25) − 125 = x³ − 15x² + 75x − 125.

What is the value of (10 − 3)³ using the identity?

(10 − 3)³ = 10³ − 3(10²)(3) + 3(10)(3²) − 3³ = 343.

How is (a − b)³ different from a³ − b³?

(a − b)³ = a³ − 3a²b + 3ab² − b³ contains two middle terms. The expression a³ − b³ has only the first and last cube terms.

How can x³ − 12x² + 48x − 64 be written as a cube?

Compare it with a³ − 3a²b + 3ab² − b³. Here a = x and b = 4, so the expression is (x − 4)³.

What is the expansion of (2a − b)³?

(2a − b)³ = 8a³ − 12a²b + 6ab² − b³.

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