elearning BlogDifference of Cubes | Formula, Factor & Examples

Difference of Cubes | Formula, Factor & Examples

Difference of cubes formula

Subtraction is an interesting concept in Mathematics. We have been finding the difference of numbers with a simple method of subtraction. With advancements in mathematical study, we encounter complex patterns of finding differences.    The difference of cubes is an expression of the subtraction of two cubes from each other. 

In this blog, you will learn the difference of cube, the cube difference formula, and how to factor perfect cubes with the help of examples. 

Difference of Cubes

The difference of cubes refers to the mathematical or algebraic expression of subtracting two perfect cubes. It aims to equate the value of the difference between two cubes. 

Difference of Cube Formula 

The difference of the cubes formula is a mathematical equation that is used to calculate the difference of numbers with the power three. The formula difference of cubes makes the factorization of cubic binomials easy.

The formula for factoring perfect cubes is expressed as:

a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2)

SOAP Math Mnemonic to Understand Cube Factoring Formula

Students usually use short tricks to remember complex formulas. SOAP Math Mnemonic is a trick that helps remember the cube factoring formula quickly. 

The letters in SOAP represent different characters. ‘S’ represents the Same sign, ‘O’ represents the Opposite sign, and ‘AP’ represents always positive. 

Let’s understand the SOAP factoring trick using the cube difference formula.
The formula

a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2)

has two parts, i.e, the left-hand side and the right-hand side. 

  • The ‘S’ or same sign refers to the sign on the first portion of the expression in R.H.S. that is the same as the sign on the left side.
  • The ‘O’ or opposite refers to the first sign in the second portion of expression in R.H.S, which will always be opposite to the sign in L.H.S. 
  • The ‘AP’ or always positive refers to the last sign in the second portion of expression in R.H.S., which will always remain positive.  
SOAP Math Mnemonic

Factoring the Difference of Cube

Factorization is the process of breaking complex equations, expressions, or numbers into their root components. The factorization of the difference of cube is carried out by using the cube factoring formula expressed as;

a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2)

Steps of Factorization of Diff of Cubes

We will follow simple steps in factorizing the diff of cubes.

  1. Find the perfect cube terms in the given expression. 
  2. Write down the formula.
  3. Convert the whole number in a cubic form if not given. 
  4. Put the values in the formula accordingly.
  5. Carry out the operation accordingly. 

Examples of Cubic Binomial Factoring

Let’s look at a few of the Cubic Binomial Factoring examples. You will learn how to factor binomials with cubes. 

Example 1: 

Factorize

x3−8x^3 - 8
  • Convert the term into a cube form.
x3−23x^3 - 2^3
8  can be written as  238 \;\text{can be written as}\; 2^3
  • Put the number in the formula.
a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2)
In our casea=xandb=2\textbf{In our case} \quad a = x \quad \text{and} \quad b = 2
x3−23=(x−2)(x2+2x+22)x^3 - 2^3 = (x - 2)(x^2 + 2x + 2^2)
=(x−2)(x2+2x+4)= (x - 2)(x^2 + 2x + 4)

So, the factors of difference of cube are

(x−2)(x2+2x+4)(x - 2)(x^2 + 2x + 4)
Therefore,x3−23=(x−2)(x2+2x+4)Therefore,x^3 - 2^3 = (x - 2)(x^2 + 2x + 4)

Example 2:

Factorize

27x3−6427x^3 - 64

using the formula for factoring perfect cubes.

  • Convert the terms into cubic form.
(3x)3−43(3x)^3 - 4^3
  • Substitute the numbers in the formula.
a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2)
a=3xandb=4a = 3x \quad \textbf{and} \quad b = 4
(3x)3−43=(3x−4)((3x)2+3x(4)+42)(3x)^3 - 4^3 = (3x - 4)( (3x)^2 + 3x(4) + 4^2 )
=(3x−4)(9x2+3x(4)+42)= (3x - 4)( 9x^2 + 3x(4) + 4^2 )
=(3x−4)(9x2+12x+16)= (3x - 4)( 9x^2 + 12x + 16 )

Example 3: 

Evaluate 343-125

  • Convert the numbers into cubic form.
73−537^3 - 5^3
  • Put the numbers in the formula.
a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2)
73−53=(7−5)(72+(7)(5)+52)7^3 - 5^3 = (7 - 5)(7^2 + (7)(5) + 5^2)
=(2)(49+35+25)= (2)(49 + 35 + 25)
=(2)(109)= (2)(109)
343−125=218343 - 125 = 218

Example 4:

Calculate

173−6317^3 - 6^3

As the cubes are given, we will simply put them in the formula and calculate them

17−6=(17−6)(172+17×6+62)17 - 6 = (17 - 6)(17^2 + 17 \times 6 + 6^2)
=(17−6)(172+17×6+62)= (17 - 6)(17^2 + 17 \times 6 + 6^2)
=(17−6)(172+102+62)= (17 - 6)(17^2 + 102 + 6^2)
=(17−6)(289+102+36)= (17 - 6)(289 + 102 + 36)
=(11)(427)= (11)(427)
173−63=(4697)17^3 - 6^3 = (4697)

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