# Expansion of (a - b)3

## Formula

• (a - b)3 = a3 - 3a2b + 3ab2 - b3.

## Notes

### Expansion of (a - b)3:

∴ (a - b)3 = (a - b)(a - b)(a - b) = (a - b)(a - b)2

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

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

= a3 - 2a2b + ab2 - a2b + 2ab2 - b3

= a3 - 3a2b + 3ab2 - b3

∴ (a - b)3 = a3 - 3a2b + 3ab2 - b3.

## Example

Expand (x - 2)3.

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

Here taking, a = x and b = 2,

(x - 2)^3 = x^3 - 3 xx x^2 xx 2 + 3 xx x xx (2)^2 - (2)^3

(x - 2)^3 = x^3 - 6x^2 + 12x - 8

## Example

Expand (4p - 5q)3

(4p - 5q)3 = (4p)3 - 3(4p)2(5q) + 3(4p)(5q)2 - (5q)3.

(4p - 5q)3 = 64p3 - 240p2q + 300pq2 - 125q3.

## Example

Find the cube of 99 using the expansion formula.

(99)3
= (100 - 1)3

= (100)3 - 3 × (100)2 × 1 + 3 × 100 × (1)2 - 13

= 1000000 - 30000 + 300 - 1

= 9,70,299

## Example

Simplify.

(2x + 3y)3 - (2x - 3y)3

(2x + 3y)3 - (2x - 3y)3

= [(2x)3 + 3(2x)2(3y) + 3(2x)(3y)2 + (3y)3 ] - [(2x)3 - 3(2x)2(3y) + 3(2x)(3y)2 - (3y)3]

= (8x3 + 36x2y + 54xy2 + 27y3) - (8x3 - 36x2y + 54xy2 - 27y3)

= 8x3 +36x2y + 54xy2 + 27y3 - 8x3 + 36x2y - 54xy2 + 27y3

= 72x2y + 54y3

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