(a – b)^3 Formula

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.

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:

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(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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