(a + b)³ Formula: A Plus B Whole Cube Formula

Mathematics becomes much easier when students understand formulas correctly. One of the most important algebraic identities is the (a + b)³ formula, also known as the A Plus B Whole Cube Formula. This formula helps students expand cubic expressions quickly and solve algebra problems efficiently.

Whether you are studying in Class 8, Class 9, Class 10, preparing for board exams, or competitive exams, understanding this identity is essential. Instead of multiplying three brackets repeatedly, you can directly apply the formula and save valuable time.

In this article, you will learn the (a + b)³ formula, its derivation, solved examples, shortcuts, common mistakes, and practice questions in a simple language.

What is the (a + b)³ Formula?

The (a + b)³ formula is an algebraic identity used to expand the cube of the sum of two numbers or variables.

The formula is:

(a + b)³ = a³ + 3a²b + 3ab² + b³

This identity is one of the most frequently used formulas in algebra.

Formula of (a − b)³

Similarly, the cube of the difference between two terms is:

(a − b)³ = a³ − 3a²b + 3ab² − b³

Students often confuse the positive and negative signs, so remember that the signs alternate in this identity.

Derivation of (a + b)³ Formula

We know:

(a + b)³

= (a + b)(a + b)(a + b)

First,

(a + b)(a + b)

= a² + 2ab + b²

Now multiply:

(a² + 2ab + b²)(a + b)

Expanding each term gives:

= a³ + a²b + 2a²b + 2ab² + ab² + b³

Combine like terms:

= a³ + 3a²b + 3ab² + b³

Therefore,

(a + b)³ = a³ + 3a²b + 3ab² + b³

Easy Trick to Remember

Simply memorize this pattern:

• First cube
• Three times first square × second
• Three times first × second square
• Last cube

Pattern:

1 : 3 : 3 : 1

This pattern makes it easy to remember the entire identity.

Solved Examples

Example 1

Expand:

(2 + 3)³

Using the formula:

= 2³ + 3×2²×3 + 3×2×3² + 3³

= 8 + 36 + 54 + 27

= 125

Example 2

Expand:

(x + 5)³

= x³ + 3x²(5) + 3x(25) + 125

= x³ + 15x² + 75x + 125

Example 3

Expand:

(3x + 2)³

= (3x)³ + 3(3x)²(2) + 3(3x)(2²) + 2³

= 27x³ + 54x² + 36x + 8

Example 4

Expand:

(a − 2)³

Using

(a − b)³

= a³ − 3a²(2) + 3a(4) − 8

= a³ − 6a² + 12a − 8

Applications of the (a + b)³ Formula

The A Plus B Whole Cube Formula is widely used in:

• Algebraic simplification
• Polynomial expansion
• Board examinations
• Competitive exams
• Engineering mathematics
• Higher algebra
• Numerical calculations
• Mental mathematics

Students who know this identity can solve many questions much faster.

Common Mistakes to Avoid

Many students make these mistakes:

• Forgetting the coefficient 3.
• Writing incorrect positive and negative signs.
• Forgetting to cube the last term.
• Mixing the formulas of (a + b)² and (a + b)³.
• Skipping intermediate calculations.

Always write every term carefully before simplifying.

Practice Questions

Try solving these yourself.

1. Expand (x + 2)³
2. Expand (a + 4)³
3. Expand (2x + y)³
4. Expand (5 + p)³
5. Expand (m − 6)³
6. Expand (3a − 2b)³

Practice regularly to improve speed and accuracy.

Why Students Should Learn Algebraic Identities

Algebraic identities are building blocks of mathematics. Once students understand formulas like (a + b)³, (a − b)³, (a + b)², and (a − b)², solving equations becomes much easier.

These formulas also improve logical thinking and problem-solving skills. They are frequently asked in school exams, Olympiads, NTSE, and other competitive examinations.

Teachers recommend learning these identities through repeated practice rather than memorization alone.

Conclusion

The (a + b)³ Formula, also known as the A Plus B Whole Cube Formula, is one of the most essential algebraic identities every student should master. By understanding its derivation, practicing solved examples, and avoiding common mistakes, students can confidently solve algebraic expressions and score better in examinations. Regular practice is the key to remembering this formula for the long term.

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