The (a – b)^3 formula is used to find the cube of a binomial which is made up of the difference of two terms. It says (a – b)3 = a3 – 3a2b + 3ab2 – b3. This is one of the algebraic identities.
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This formula is used to calculate the cube of the difference between two terms very easily and quickly without doing complicated calculations. Let us learn more about a minus b whole cube formula along with solved examples.
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What is the (a – b)^3 Formula?
The (a-b)^3 formula is used to calculate the cube of a binomial. The formula is also known as the cube of the difference between two terms. According to “a minus b whole cube formula”,
(a – b)3 = a3 – 3a2b + 3ab2 – b3 (or) a3 – b3 – 3ab (a – b)
We can derive this formula in two ways:
- Method 1: By expanding (a – b)3 as (a – b) (a – b) (a – b).
- Method 2: By using the formula of (a + b)3
Derivation of (a – b)^3: Method 1
To find the formula of (a – b)3, we will just multiply (a – b) (a – b) (a – b).
(a – b)3 = (a – b)(a – b)(a – b)
= (a2 – 2ab + b2)(a – b)
= a3 – a2b – 2a2b + 2ab2 + ab2 – b3
= a3 – 3a2b + 3ab2 – b3 (or)
= a3 – b3 – 3ab (a – b)
Therefore, (a – b)3 formula is:
(a – b)3 = a3 – 3a2b + 3ab2 – b3
Hence proved.
Derivation of (a – b)^3: Method 2
We use the formula of (a + b)3 to derive the formula of a minus b whole cube. We know that
(a + b)3 = a3 + 3a2b + 3ab2 + b3
Replace b with -b on both sides of this formula:
(a + (-b))3 = a3 + 3a2(-b) + 3a(-b)2 + (-b)3
This results in (a – b)3 = a3 – 3a2b + 3ab2 – b3.
Hence derived.
Examples on (a – b)^3 Formula
Example 1: Solve the following expression using (a – b)3 formula: (2x – 3y)3
Solution:
To find: (2x – 3y)3
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Using (a – b)3 Formula,
(a – b)3 = a3 – 3a2b + 3ab2 – b3
= (2x)3 – 3 × (2x)2 × 3y + 3 × (2x) × (3y)2 – (3y)3
= 8×3 – 36x2y + 54xy2 – 27y3
Answer: (2x – 3y)3 = 8×3 – 36x2y + 54xy2 – 27y3
Example 2: Find the value of x3 – y3 if x – y = 5 and xy = 2 using (a – b)3 formula.
Solution:
To find: x3 – y3
Given:
x – y = 5
xy = 2
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Using (a – b)3 Formula,
(a – b)3 = a3 – 3a2b + 3ab2 – b3
Here, a = x; b = y
Therefore,
(x – y)3 = x3 – 3 × x2 × y + 3 × x × y2 – y3
(x – y)3 = x3 – 3x2y + 3xy2 – y3
53 = x3 – 3xy(x – y) – y3
125 = x3 – 3 × 2 × 5 – y3
x3 – y3 = 155
Answer: x3 – y3 = 155
Example 3: Solve the following expression using (a – b)3 formula: (5x – 2y)3
Solution:
To find: (5x – 2y)3
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Using (a – b)3 Formula,
(a – b)3 = a3 – 3a2b + 3ab2 – b3
= (5x)3 – 3 × (5x)2 × 2y + 3 × (5x) × (2y)2 – (2y)3
= 125×3 – 150x2y + 60xy2 – 8y3
Answer: (5x – 2y)3 = 125×3 – 150x2y + 60xy2 – 8y3
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