# $(A+B)^{3}$ Formula – Solved Examples, FAQS, Practice Problems

Home » Math Vocabulary » $(A+B)^{3}$ Formula – Solved Examples, FAQS, Practice Problems

## What Is the (a + b)3 Formula?

In algebra, the (a + b)3 formula allows us to find the cube of a binomial and helps us to simplify expressions

The formula for the cube of the sum of two terms, (a + b)3, can be expanded using the binomial theorem or by applying the distributive property multiple times. The result is:

(a + b)3 = a3 + 3a2b + 3 ab2 + b3

This formula shows how to find the cube of the sum of a and b by raising both terms to the power of 3, along with the combinations involving a and b raised to the power of 2 and 1.

## (a + b)3 Formula

The (a + b) whole cube formula or (a + b)3 formula represents the cube of a binomial. It represents the result of cubing the sum of ‘a’ and ‘b’.

(a + b)3 = a3 + 3a2b + 3ab2 + b3

It’s important to note that this formula holds true for any values assigned to both ‘a’ and ‘b.’

## Derivation of (a+b)3 Formula

To understand the derivation of the a + b whole cube formula, let’s start by expanding the expression (a + b)3

We can achieve this by multiplying the binomial (a + b) by itself three times:

(a + b)3 =  (a + b)(a + b)(a + b)

On expanding this expression, we get

(a + b)3 = (a2 + 2ab + b2)(a + b)

Let’s use the distributive property to expand the above expression.

(a + b)3 = a3 + a2b + 2a2b + 2ab2 + ab2 + b3

On combining like terms, we arrive at the final form of the (a + b)3 formula:

(a + b)3 = a3 + 3a2b + 3ab2 + b3

Alternatively, we can express the (a + b)3 formula as:

(a + b)3 = a3 + 3ab (a + b) + b3

This derivation showcases the algebraic manipulation required to arrive at the (a + b)3 formula.

## How to Use the (a+b)3 Formula

Let’s use the formula (a + b)3 = a3 + 3a2b + 3ab2 + b3 to expand some expressions.

Example: (x + 2y)3

Here, a = x, b = 2x

(a + b)3 = a3 + 3a2b + 3ab2 + b3

(x + 2y)3 = x3 + 3x2 (2y) + 3x(2y)2 + (2y)3

(x + 2y)3 = x3 + 6x2y + 3x(4y2) + 8y3

(x + 2y)3 = x3 + 6x2y +12xy2 + 8y3

## Facts about the (a+b)3 Formula

• The formula (a + b)3 represents the cube of the sum of two terms, “a” and “b.” (a + b)3 = a3 + 3a2b + 3ab2 + b3
• The a plus b whole cube formula represents the cube of the sum of “a” and “b.” It can also be expressed as (a + b)3 = (a + b)$\times$(a + b)$\times$(a + b).
• The (a + b)3 formula is part of a group of related formulas known as the binomial coefficients, which are used to expand powers of binomials.

For example, (a + b)2 = a2 + 2ab + b2

• The coefficients of the terms in the expansion of (a + b)3 correspond to the numbers in the fourth row of Pascal’s Triangle. Pascal’s Triangle is a triangular arrangement of numbers where each number is the sum of the two numbers directly above it.

In the expansion (a + b)3 = a3 + 3a2b + 3ab2 + b3, the coefficients (1, 3, 3, 1) are the same as the fourth row of Pascal’s Triangle. This relationship between binomial expansions and Pascal’s Triangle extends to higher powers as well.

## Conclusion

The (a+b)3 formula is a powerful tool in algebra for expanding and simplifying cubic binomials. By understanding its derivation, properties, and applications, you can enhance your problem-solving skills and tackle a wide range of mathematical challenges. Let’s solve some  (a + b)3 formula examples and practice problems to understand the concept better.

## Solved Examples on the (a+b)3 Formula

Example 1. Expand (x + 2)3.

Solution:

Using the (a + b)3  formula, we have:

(a + b)3 = a3 + 3a2b + 3ab2 + b3

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

Simplifying each term, we get

(x + 2)3 = x3 + 6x2 + 12x + 8

Example 2: Evaluate 143 using the (a + b)3  formula.

Solution:

Hare, we can write

143 = (10 + 4)3

Here, a = 10, b = 4

(a + b)3 = a3 + 3a2b + 3ab2 + b3

(10 + 4)3 = 103 + 3(10)2(4) + 3(10)(4)2 + (4)3

(10 + 4)3 = 1000 + 3(100)(4) + 3(10)(16) + 64

(10 + 4)3 = 1000 + 1200 + 480 + 64

(10 + 4)3 = 2744

143 = 2744

Example 3: Find the coefficient of the term a²b in the expansion of (3a + 2b)³.

Solution:

Using the (a + b)3 formula, we get

(3a + 2b)3 = (3a)3 + 3(3a)2(2b) + 3(3a)(2b)2 + (2b)3

(3a + 2b)3 = 27a3 + 3(9a2)2b + 3(3a)(4b2) + 8b3

(3a + 2b)3 = 27a3 + 54a2b +36ab2 + 8b3

So, the coefficient of the term involving “a²b” is 54.

## Practice Problems on the (a + b)3 Formula

1

### What is the expansion of $(a + b)^{3}$?

$a^{3} + b^{3}$
$a^{3} + 3a^{2}b + 3ab^{2} + b^{3}$
$(a^{2} + 2ab + b^{2})$
$a^{3} + 2a^{2}b + 2ab^{2} + b^{3}$
CorrectIncorrect
Correct answer is: $a^{3} + 3a^{2}b + 3ab^{2} + b^{3}$
The correct expansion of $(a + b)^{3}$ is $a^{3} + 3a^{2}b + 3ab^{2} + b^{3}$.
2

### What does $(a + b)^{3}$ represent?

Square of the sum of a and b
Cube of the sum of a and b
Sum of cubes of a and b
Product of a and b raised to the power of 3
CorrectIncorrect
Correct answer is: Cube of the sum of a and b
The $(a + b)^{3}$ formula represents the cube of the sum of a and b
3

### Which pattern do the coefficients in the expansion of $(a + b)^{3}$ follow?

Sequential increase
Pascal's triangle arrangement
Fibonacci number sequence
CorrectIncorrect
Correct answer is: Pascal's triangle arrangement
The coefficients in the expansion of $(a + b)^{3}$ follow the arrangement of Pascal's triangle, where each coefficient is obtained by adding the two coefficients directly above it.
4

### What is the expanded form of $(x + 2y)^{3}$?

$x^{3} + 2y^{3}$
$x^{3} + 6x^{2}y + 12xy^{2} + 8y^{3}$
$x^{3} + 3x^{2}y + 3xy^{2} + 2y^{3}$
$x^{3} + 3xy$
CorrectIncorrect
Correct answer is: $x^{3} + 6x^{2}y + 12xy^{2} + 8y^{3}$
The expansion of $(x + 2y)^{3}$ is obtained by applying the $(a + b)^{3}$ formula, resulting in $x^{3} + 6x^{2}y + 12xy^{2} + 8y^{3}$