# Cube of the Difference of Two Binomials

What is the formula for the cube of the difference of two binomials?

To determine cube of a number means multiplying a number with itself three times similarly, cube of a binomial means multiplying a binomial with itself three times.

(a - b) (a - b) (a - b) = (a - b)3

or, (a - b) (a - b) (a - b) = (a - b) (a - b)2

= (a – b) (a2 + b2 - 2ab),
[Using the formula of (a + b) 2 = a2 - 2ab + b2]

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

= a3 + ab2 – 2a2b – ba2 – b3 + 2ab2

= a3 – 3a2b + 3ab2 – b3

Therefore, (a - b)3 = a3 – 3a2b + 3ab2 – b3

Thus, we can write it as; a = first term, b = second term

(First term – Second term)3 = (first term)3 - 3 (first term)2 (second term) + 3 (first term) (second term)2 - (second term)3

So, the formula for the cube of the difference of two terms is written as:

(a - b)3 = a3 – 3a2b + 3ab2 – b3

= a3 – b3 – 3ab (a - b)

Worked-out examples to find the cube of the difference of two binomials:

1. Determine the expansion of (3x – 4y)3

Solution:

We know, (a - b)3 = a3 – 3a2b + 3ab2 – b3

(3x – 4y)3

Here, a = 3x, b = 4y

= (3x)3 – 3 (3x)2 (4y) + 3 (3x) (4y)2 – (4y)3

= 27x3 – 3 (9x2) (4y) + 3 (3x) (16y2) – 64y3

= 27x3 – 108x2y + 144xy2 – 64y3

Therefore, (3x – 4y)3 = 27x3 – 108x2y + 144xy2 – 64y3

2. Use the formula and evaluate (997)3

Solution:

(997)3 = (1000 – 3)3

We know, (a - b)3 = a3 – 3a2b + 3ab2 – b3

Here, a = 1000, b = 3

(1000 – 3)3

= (1000)3 – 3 (1000)2 (3) + 3 (1000) (3)2 – (3)3

= 1000000000 – 9 (1000000) + (3000) 9 – 27

= 1000000000 – 9000000 + 27000 – 27

= 991026973

Therefore, (997)3 = 991026973

Thus, to expand the cube of the difference of two binomials we can use the formula to evaluate.

Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

## Recent Articles

1. ### Types of Fractions |Proper Fraction |Improper Fraction |Mixed Fraction

Mar 02, 24 05:31 PM

The three types of fractions are : Proper fraction, Improper fraction, Mixed fraction, Proper fraction: Fractions whose numerators are less than the denominators are called proper fractions. (Numerato…

2. ### Subtraction of Fractions having the Same Denominator | Like Fractions

Mar 02, 24 04:36 PM

To find the difference between like fractions we subtract the smaller numerator from the greater numerator. In subtraction of fractions having the same denominator, we just need to subtract the numera…

3. ### Addition of Like Fractions | Examples | Worksheet | Answer | Fractions

Mar 02, 24 03:32 PM

To add two or more like fractions we simplify add their numerators. The denominator remains same. Thus, to add the fractions with the same denominator, we simply add their numerators and write the com…

4. ### Comparison of Unlike Fractions | Compare Unlike Fractions | Examples

Mar 01, 24 01:42 PM

In comparison of unlike fractions, we change the unlike fractions to like fractions and then compare. To compare two fractions with different numerators and different denominators, we multiply by a nu…