Sum of Cubes Formula, Definition, Proof, Examples

Sum of Cubes Formula: The mathematical expression used to calculate the sum of two polynomials, represented as a³ + b³, is termed the "sum of cubes formula." Exploring this formula through solved examples proves quite insightful. This factoring method becomes extremely useful when dealing with diverse algebraic expressions. Memorizing this formula is simple and achievable within a few minutes. It shares similarities with the difference of the cubes formula.

Understanding the Sum of Cubes Formula

In this section, we will discuss deeper into the concept of the sum of cubes. The formula is expressed as:

a³ + b³ = (a + b)(a² - ab + b²)

Here, we define the variables as follows:

'a' represents the first variable.

'b' stands for the second variable

Proof of the Sum of Cubes Formula

To validate the  formula, a ³ + b ³ = (a + b) (a ² - ab + b ² )

We aim to demonstrate LHS = RHS.

The left-hand side (LHS) term is a ³ + b ³

Upon solving the right-hand side (RHS) term, we acquire: (a + b) (a ² - ab + b ² )

By multiplying 'a' and 'b' separately with (a ² - ab + b ² )

= a (a ² - ab + b ² ) + b(a ² - ab + b ² )

= a ³ - a ² b + ab ² + a ² b - ab ² + b ³

= a ³ - a ² b + a ² b + ab ² - ab ² + b ³

= a ³ - 0 + 0 + b ³

= a ³ + b ³

Therefore, it is verified that LHS = RHS. Hence, the proof confirms the equality of both sides of the equation.

Sum of Cubes Formula Solved Examples

Example 1: Applying the sum of cubes formula to factor 216x³ + 64.

To determine the factor of 216x³ + 64 using the sum of cubes formula: 216x ³ +64=(6x) ³ +4 ³

Using the  formula: a ³ +b ³ =(a+b)(a ² −ab+b ² )

Substituting the values:

(6x) ³ +4 ³ =(6x+4)((6x) ² −(6x)(4)+4 ² )

(6x) ³ +4 ³ =(6x+4)(36x ² −24x+16)

(6x) ³ +4 ³ =8(3x+2)(9x ² −6x+4)

Answer: The factor of 216x ³ +64 is 8(3x+2)(9x ² −6x+4).

Example 2: Factoring 8x³ + 125y³ using the sum of cubes formula.

To find the factor of 8x ³ +125y ³ applying the formula:

8x ³ +125y ³ =(2x) ³ +(5y) ³

Using the formula:

a ³ +b ³ =(a+b)(a ² −ab+b ² )

Substituting the values:

(2x) ³ +(5y) ³ =(2x+5y)((2x) ² −(2x)(5y)+(5y) ² )

(2x) ³ +(5y) ³ =(2x+5y)(4x ² −10xy+25y ² )

Example 3: Factorize 27x³ - 8y³ using the sum of cubes formula.

To factorize 27x ³ −8y ³ using the formula:

27x ³ −8y ³ =(3x) ³ −(2y) ³

Using the sum of cubes formula: a ³ −b ³ =(a−b)(a ² +ab+b ² )

Substituting the values:

(3x) ³ −(2y) ³ =(3x−2y)((3x) ² +(3x)(2y)+(2y) ² )

(3x) ³ −(2y) ³ =(3x−2y)(9x ² +6xy+4y ² )

Example 4: Factor the expression 64a³ - 125b³ using the sum of cubes formula.

To factorize 64a ³ −125b ³ applying the sum of cubes formula:

64a ³ −125b ³ =(4a) ³ −(5b) ³

Using the  formula: a ³ −b ³ =(a−b)(a ² +ab+b ² )

Substituting the values:

(4a) ³ −(5b) ³ =(4a−5b)((4a) ² +(4a)(5b)+(5b) ² )

(4a) ³ −(5b) ³ =(4a−5b)(16a ² +20ab+25b ² )

These examples demonstrate the application of the sum of cubes formula to factorize expressions into manageable forms using the relevant algebraic identity.

Sum of Cubes Formula a ³ +b ³ =(a+b)(a ² −ab+b ² ) serves as a valuable concept in algebra, allowing efficient factorization of expressions involving the sum of two cube terms. Its application simplifies problem-solving in various mathematical contexts, enabling quicker manipulations and solutions in algebraic equations and expressions.

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