The sum and difference of cubes are algebraic formulas used to factor expressions of the form a³+b³ and a³−b³ respectively. These formulas are particularly useful in simplifying and solving polynomial equations.

It is the basic formula of algebra used to solve the sum of the cubes and the difference of the cubes without actually calculating the values of the cubes. The sum of the cubes of the polynomial is represented as, a³ + b³ whereas the difference of the cubes is represented as a³– b³. These algebraic expressions are easily factorized using various algebraic expressions without actually calculating the cubes.

In this article, we will learn about Sum of Cubes, Sum of Cubes Formula, Factoring Sum of Cubes, Difference of Cubes, Difference of Cubes Formula with examples in detail below.

Sum and Difference of Cubes

Sum of Cubes

Sum of cubes is the formula that is used to find the sum of two cubes without actually finding their cubes arithmetically. The sum of cubes is very useful in solving various algebraic problems and is helpful in quickly solving various problems. The sum of cubes formula is the formula that is used to factorize the sum of cubes, The formula for the sum of cubes is discussed below:

Sum of Cubes Formula

Cube of a number is the number multiplied by itself twice. The sum of the cube is the formula that is the formula for a³ + b³ and its formula is added below,

a³ + b³ = (a + b)(a² – ab + b²)

The above formula is algebraic identity and that is used to find the sum of cubes formula.

Sum of Cube Formula Proof

This identity can be proved by multiplying the expressions on the right side and getting equal to the left side expression. Here is the proof of this identity.

Given Identity:

a³ + b³ = (a + b) (a² – ab + b²)

Proof:

= RHS

= (a + b)(a² – ab + b²)

= a(a² – ab + b²)) + b(a² – ab + b²)

= a³ – a²b + ab² + a²b – ab² + b³

= a³ – a²b + a²b + ab² – ab² + b³

= a³ + b³ 

= LHS

Hence proved.

Factoring Sum of Cubes

We use the sum of cubes formula to easily factorize the cubes in polynomials. This is explained by the example added below,

For example, suppose we have to factorize, x³ + 27

Solution:

= x³ + 27

= x³ + 3³

Using Identity, a³ + b³ = (a + b) (a² – ab + b²)

where,

• a = x
• b = 3

= (x + 3)(x² -(x)(3) + 3²)

= (x + 3)(x² – 3x + 9)

Thus, the factors of x³ + 27 are easily found.

Difference of Cubes

When subtracting any two polynomials, a³ – b³, the difference of cubes formula is utilized. This formula is easy to memorize and may be completed in minutes. It is similar to how the sum of cubes formula works. 

Difference of Cube Formula

a³ – b³ = (a – b) (a² + ab + b²)

Difference of Cube Formula Proof

This identity can be proved by multiplying the expressions on the right side and getting equal to the left side expression. Here is the proof of this identity.

Given Identity:

a³ – b³ = (a – b) (a² + ab + b²)

Proof:

= RHS 

= (a – b)(a² + ab + b²)

= a(a² + ab + b²)) – b(a² + ab + b²)

= a³ + a²b + ab² – a²b – ab² – b3

= a3 – a²b + a²b + ab² – ab² – b3

= a³ – b³

= LHS

Hence proved.

Factoring Difference of Cubes

We use the difference of cubes formula to easily factorize the cubes in polynomials. This is explained by the example added below:

For example, suppose we have to factorize, x³ – 343

Solution:

= x³ – 343

= x³ – 7³

Using identity a³ – b³ = (a – b) (a² + ab + b²)

where,

• a = x
• b = 7

= (x – 7) (x² + (x)(7) + 7²)

= (x – 7) (x² + 7x + 49)

Thus, the factors of x³ – 343 are easily found.

Read More

• Algebraic Expressions
• Factorization of Polynomial
• Polynomial Formula

Examples on Sum and Difference of Cubes

Example 1: Factorize y³ – 125

Solution:

y³ – 125 = y³ – 5³

Since, a³ – b³ = (a – b) (a² + ab + b²), 

here,

• a = y
• b = 5

= (y – 5) (y² + (y)(5) + 5²)

= (y – 5) (y² + 5y + 25)

Example 2: Evaluate 25³ – 12³

Solution:

Since, a³ – b³ = (a – b) (a² + ab + b²), 

where,

• a = 25 
• b = 12

= 25³ – 12³ 

= (25 – 12) (25² + (25)(12) + 12²)

= 13 (625 + 300 + 144)

= 13897

Example 3: Factorize 8p³ + 27

Solution:

8p³ + 27 = (2p)³ + (3)³

Since, a³ + b³ = (a + b)(a² – ab + b²)

= (2p)³ + (3)³ 

= (2p + 3)[(2p)² – (2p)(3) + (3)²]

= (2p + 3)[4p² – 6p + 9]

Example 4: Factorize 512 + 729v³

Solution:

512 + 729v³ = (8)³ + (9v)³

Since, a³ + b³ = (a + b)(a² – ab + b²)

= (8)³ + (9v)³ 

= (8 + 9v)[(8)² – (8)(9v) + (9v)²]

= (8 + 9v)[64 – 72v + 729v²]

Example 5: Solve: 25³ + 12³

Solution:

Since, a³ + b³ = (a + b) (a² – ab + b²)

where,

• a = 25
• b = 12

= 25³ + 12³ 

= (25 + 12) (25² – (25)(12) + 12²)

= 37 (625 – 300 + 144)

= 17353

Practice Problems on Sum and Differences of Cubes

Q1. Factorize 64 + 343v³

Q2. Factorize 64 – 343v³

Q3. Evaluate 15³ – 9³

Q4. Evaluate 23³ – 7³

Sum and Differences of Cubes – FAQs

What is the sum of cubes?

Sum of the cubes is the used to find the sum of two cubes without actually finding its cubes arithmetically. It is represented as a³ + b³.

What is the sum of cubes formula?

The sum of the cubes formula is the formula that is used to factorize the a³ + b³ and its formula is, a³ + b³ = (a + b) (a² – ab + b²)

What is the difference of cubes?

Difference of the cubes is the used to find the difference of two cubes without actually finding its cubes arithmetically. It is represented as a³ – b³.

What is the difference of cubes formula?

The difference of the cubes formula is the formula that is used to factorize the a³ – b³ and its formula is, a³ – b³ = (a – b) (a² + ab + b²)

What is the cube of 9?

The cube of 9 is 729, i.e. 9³ = 9×9×9 = 729.

What is the cube root of 1331?

The cube root of 1331 is 11, i.e. ∛(1331) = ∛(11×11×11) = 11.

What is the cube of 13?

The cube of 13 is 729, i.e. 13³ = 13×13×13 = 2197.

What is cube root of 8?

The cube root of 8 is 2, i.e. ∛(8) = ∛(2×2×2) = 2.