Difference of Cubes

Difference of Cubes is the formula in mathematics that is used to simplify the difference between two cubes. This formula is used to solve the difference of cubes without actually finding the cubes. This formula factorizes the difference of a cube and changes it into other forms. The difference of cube is also called the a³ b³ formula or the a³ – b³ formula.

In this article, we have covered the difference of cubes, the difference of cube formulas, various examples related to that formula, and others in detail.

What is Difference of Cubes?

Difference of cubes is the basic formula in mathematics that is used to find the difference between the cubes without actually calculating the cubes. The difference of cubes is very useful in solving various polynomial problems, and simplifying algebraic problems. It is the standard algebraic identity that is used in mathematics. The difference of cubes formula is the formula that is used to calculate the differences of the cube and the same formula is added below.

Difference of Cubes Formula

A cube is a number multiplied by itself twice. The difference of cube formula is the formula that is used to find difference of two cubes without actually finding the cube. Suppose we have two numbers a and b then their cubes are a³and b³. Now the difference of their cubes would be, a³ – b³

In mathematics, the formula for the expression a³ – b³ is,

a³ – b³ = (a – b) (a² + ab + b²)

It is an algebraic identity that is used to find the difference between two cubes without actually calculating the cubes. To factorize the binomials of cubes, the difference of cubes formula is used.

Derivation of Difference of Cube Formula

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 – b³

= LHS

Thus, identity is proved.

Factoring Cubes Formula

We use the difference of cubes formula to easily factorize the cubes in polynomial. 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.

Read More,

• Algebraic Expressions
• Factorization of Polynomial
• Polynomial Formula

Examples on Difference of Cubes

Example 1: Evaluate 12³ – 8³.

Solution:

Use the identity,

a³ – b³ = (a – b) (a² + ab + b²)

where,

• a = 12
• b = 8

= 12³ – 8³

= (12 – 8) (12² + (12)(8) + 8²)

= 4 (144 + 96 + 64)

= 4 (304)

= 1216

Example 2: Evaluate 15³ – 10³.

Solution:

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

where,

• a = 15
• b = 10

15³ – 10³

= (15 – 10) (15² + (15)(10) + 10²)

= 5 (225 + 150 + 100)

= 5 (475)

= 2375

Example 3: Evaluate 19³ – 9³.

Solution:

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

where,

• a = 19
• b = 9

= 19³ – 9³

= (19 – 9) (19² + (19)(9) + 9²)

= 10 (361 + 171 + 81)

= 5 (613)

= 3065

Example 4: Factorize x³ – 343.

Solution:

x³ – 343 = x³ – 7³

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

where,

• a = x
• b = 7

= (x – 7) (x² + (x)(7) + 7²)

= (x – 7) (x² + 7x + 49)

Example 5: Factorize y³ – 125.

Solution:

y³ – 125 = y³ – 5³

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

where,

• a = y
• b = 5

= (y – 5) (y² + (y)(5) + 5²)

= (y – 5) (y² + 5y + 25)

Example 6: Factorize x⁹ – 512.

Solution:

x⁹ – 512 = (x³)³ – 8³

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

where,

• a = x³
• b = 8

= (x³ – 8) ((x³)² + (x³)(8) + 8²)

= (x³ – 2³) (x⁶ + 8x³ + 64)

Again using the identity a³ – b³ = (a – b) (a² + ab + b²)

where,

• a = x
• b = 2

= (x – 2) (x² + (x)(2) + 2²) (x⁶ + 8x³ + 64)

= (x – 2) (x² + 2x + 4) (x⁶ + 8x³ + 64)

Practice Problems on Difference of Cubes

P1. Factorize a⁶ – 64

P2. Factorize 27x³ – 1331

P3. Evaluate 21³ – 11³

P4. Evaluate 13³ – 9³

FAQs on Difference of Cubes

1. What is Difference of Cubes?

Difference of cubes is the method of calculating the differences of two cubes without actually calculating the cubes. The difference of cubes actually uses algebraic identities to find the required answer.

2. What is Difference of Cube Formula?

The difference of cube formula is the formula that is used to calculate the difference of two cubes. The difference of cube formula in mathematics is,

a³ – b³ = (a – b) (a² + ab + b²)

3. What is Formula for a³ – b³?

The formula for a³ – b³ is also called the difference of cubes formula and it is given as,

a³ – b³ = (a – b) (a² + ab + b²)

4. What is Cube of 3?

Cube of 3 is 27, i.e. 3³ = 3×3×3 = 27

5. What is Cube of 2?

Cube of 2 is 8, i.e. 2³ = 2×2×2 = 8

6. What is Cube of 11?

Cube of 11 is 1331, i.e. 11³ = 11×11×11 = 1331