The a-b whole cube formula, i.e. (a-b)3 formula, is used to find the cube of the difference between two terms. This formula is also used to factorise some types of trinomials. The a-b whole cube formula is one of the important algebraic identities. Generally, the (a-b)3 formula is used to solve the problems quickly without undergoing any complicated calculations. In this article, we are going to learn the a-b whole cube formula, derivation and examples in detail.
A-B Whole Cube Formula
(a-b)3 formula is used to calculate the cube of a binomial. (a-b)3 is nothing but (a-b)(a-b)(a-b).
The a-b whole cube formula is given by:
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(a – b)3 = a3 – 3a2b + 3ab2 – b3 |
(a-b)^3 Formula Derivation
To derive the formula for (a-b)3, we have to multiply (a-b) thrice by itself. (i.e) (a-b)(a-b)(a-b). Go through the below steps to find the formula for (a-b)3.
Derivation:
(a-b)3 = (a-b)(a-b)(a-b)
(a-b)3 = (a2-2ab+b2) (a-b) [Since, (a-b)2 = a2+b2-2ab)
(a-b)3 = a3-2a2b+ab2-a2b+2ab2-b3
(a-b)3 = a3-3a2b+3ab2– b3
Therefore, the formula for (a-b)3 is a3-3a2b+3ab2– b3.
The above formula can also be written as:
(a-b)3 = a3-3ab(a-b) – b3.
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Examples on (a-b)^3
Example 1:
Solve the expression (x-2y)3.
Solution:
Given expression: (x-2y)3.
We know that (a-b)3 = a3-3a2b+3ab2– b3
In the expression (x-2y)3, a = x and b = 2y.
Now, substitute the value in the a-b whole cube formula, we get
(x-2y)3 = x3– 3(x)2(2y) + 3(x)(2y)2 – (2y)3
(x-2y)3 = x3 – 6x2y+12xy2 – 8y3.
Hence, (x-2y)3 = x3 – 6x2y+6xy2 – 8y3.
Example 2:
Solve the expression: (2x – 7y)3
Solution:
Given: (2x – 7y)3.
As we know, (a-b)3 = a3-3a2b+3ab2– b3
Here, a = 2x and b = 7y
By substituting the values in the algebraic identity, we get
(2x – 7y)3 = (2x)3 – 3(2x)2(7y) + 3(2x)(7y)2 – (7y)3
(2x -7y)3 = 8x3 – 84x2y +294xy2 – 343y3
Therefore, (2x – 7y)3 = 8x3 – 84x2y +294xy2 – 343y3.
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