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

The (a + b)³ formula, also called the a plus b whole cube formula, is used to expand the cube of the sum of two terms. The formula is:

(a + b)³ = a³ + 3a²b + 3ab² + b³

It helps students expand algebraic expressions quickly without multiplying the same binomial three times.

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(a + b)³ Formula

The (a + b)³ formula is:

(a + b)³ = a³ + 3a²b + 3ab² + b³

Here:

• a³ is the cube of the first term.

• 3a²b is the second term.

• 3ab² is the third term.

• b³ is the cube of the second term.

The coefficients in the expansion are 1, 3, 3, 1.

A Plus B Whole Cube Formula

The a plus b whole cube formula is another way of writing the same identity:

(a + b)³ = a³ + 3a²b + 3ab² + b³

For example:

(x + 2)³ = x³ + 6x² + 12x + 8

This identity is useful when expanding algebraic expressions containing a binomial raised to the power of 3. CuriousJr helps students understand and apply this identity through simple examples.

(a + b)³ Formula Expansion

To understand the expansion, start with:

(a + b)³ = (a + b)(a + b)(a + b)

First multiply two brackets:

(a + b)(a + b) = a² + 2ab + b²

Now multiply by the third bracket:

(a² + 2ab + b²)(a + b)

Expanding and combining like terms gives:

a³ + a²b + 2a²b + 2ab² + ab² + b³

Therefore:

(a + b)³ = a³ + 3a²b + 3ab² + b³

How to Use the (a + b)³ Formula

Follow these steps to use the formula:

1. Identify the values or terms represented by a and b.

2. Write the identity (a + b)³ = a³ + 3a²b + 3ab² + b³.

3. Substitute the values of a and b.

4. Calculate each term.

5. Combine the terms to get the final expression.

Example 1: Expand (2 + 3)³

Here, a = 2 and b = 3.

Using the formula:

(2 + 3)³ = 2³ + 3(2²)(3) + 3(2)(3²) + 3³

= 8 + 36 + 54 + 27

= 125

Therefore:

(2 + 3)³ = 125

Example 2: Expand (x + 2)³

Here, a = x and b = 2.

Using the formula:

(x + 2)³ = x³ + 3(x²)(2) + 3(x)(2²) + 2³

= x³ + 6x² + 12x + 8

Therefore:

(x + 2)³ = x³ + 6x² + 12x + 8

Example 3: Expand (3 + y)³

Here, a = 3 and b = y.

Using the formula:

(3 + y)³ = 3³ + 3(3²)(y) + 3(3)(y²) + y³

= 27 + 27y + 9y² + y³

Therefore:

(3 + y)³ = 27 + 27y + 9y² + y³

Read More: a^2 + b^2 Formula

What Is the Formula of (a + b)³?

The formula of (a + b)³ is:

(a + b)³ = a³ + 3a²b + 3ab² + b³

It represents the cube of the sum of a and b. The four terms in the expansion contain the cubes of the individual terms and the two middle terms formed using both a and b.

(a + b)³ Formula Coefficients

The coefficients of the expansion are:

So, the coefficient pattern is:

1, 3, 3, 1

The powers of a decrease from 3 to 0, while the powers of b increase from 0 to 3.

(a + b)³ and (a - b)³ Formulas

The GSC data also shows strong searches for the related (a - b)³ identity. It is useful to know the difference between the two formulas.

(a + b)³

(a + b)³ = a³ + 3a²b + 3ab² + b³

All terms in the expansion are positive when a and b are positive quantities.

(a - b)³

(a - b)³ = a³ - 3a²b + 3ab² - b³

The signs alternate in this expansion.

Do not confuse the two identities while solving algebraic expressions.

Difference Between (a + b)³ and (a - b)³

The main difference is the signs of the terms. For (a + b)³, the expansion contains positive terms. For (a - b)³, the signs alternate.

A Cube Plus B Cube

The cube of a sum should not be confused with the sum of two cubes.

For the cube of a sum:

(a + b)³ = a³ + 3a²b + 3ab² + b³

For the sum of cubes:

a³ + b³ = (a + b)(a² - ab + b²)

These are different algebraic identities and are used in different types of problems.

How to Remember the (a + b)³ Formula

Use this pattern to recall the identity:

First cube + 3 × first square × second + 3 × first × second square + second cube

So:

(a + b)³

becomes

a³ + 3a²b + 3ab² + b³

A quick way to check the expansion is to remember the coefficient pattern:

1 → 3 → 3 → 1

Also check that the total power of a and b in each term is 3.

Read More: A^2-B^2 Formula

Common Mistakes in the (a + b)³ Formula

Students often make these mistakes while using the whole cube formula:

• Writing a³ + b³ as the complete expansion of (a + b)³.

• Forgetting the coefficient 3 in the middle terms.

• Writing 3a²b², which has total power 4 instead of 3.

• Confusing the signs in (a + b)³ and (a - b)³.

• Leaving out the final term b³.

For example, the incorrect expansion:

(a + b)³ = a³ + b³

misses the two middle terms.

The correct expansion is:

(a + b)³ = a³ + 3a²b + 3ab² + b³

Quick Revision of (a + b)³ Formula