Question

what is the formula of (a-b)³.
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Answer 1

Answer:

The formula is (a-b)³=a³-3a²b+3ab²-b³. You convert it to (a-b)³=a³-3ab(a-b)-b³ for an illustration. You take the drawing of the formula (a+b)³=a³+3a²b+3ab²+b³ from above and replace a by the difference a-b. Then the edges are (a-b)+b with different combinations (on the left).

= (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

Or

= a³-b³-3a²b+3ab²

= a³-b³-3ab(a-b)

Therefore,

(a-b)³ = a³-3a²b+3ab²-b³

Or

(a-b)³ = a³-b³-3ab(a-b)

•••

Step-by-step explanation:

Answer 2

Identity :-

(a-b)³ = a³ - 3a²b + 3ab² - b³

(a-b)³ = a³ - 3ab(a-b) - b³

Examples :-

Let's verify this identity with some examples :

(4-2)³

a = 4 and b = 2

(4-2)³ = 4³ - 3(4)(2)(4-2) - 2³

(2)³ = 64 - 48 - 8

(2)³ = 64 - 56

(2)³ = 8

LHS = RHS

Hence verified.

(1-2)³

where a = 1 and b = 2

(1-2)³ = (1)³ - 3(1)(2)(1-2) - (2)³

(-1)³ = 1 - 6(-1) - 8

(-1)³ = 1 + 6 - 8

(-1)³ = 1 - 2

(-1)³ = (-1)

LHS = RHS

Hence verified.

Let's solve with variables in the question :

1. (3a-b)³
2. (7x-4)³
3. (12pq-3q)

(3a-b)³

a = 3a and b = b

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

(3a-b)³ = (3a)³ - 9ab(3a-b) - (b)³

(3a-b)³ = 27a³ - 27a²b + 9ab² - b³

(7x-4)³

a = 7x and b = 4

(7x-4)³ = (7x)³ - 3(7x)(4)(7x-4) - (4)³

(7x-4)³ = (7x)³ - 84x(7x-4) - (4)³

(7x-4)³ = 343x³ - 588x² + 336x - 64

(12pq-3q)³

a = 12pq and b = 3q

(12pq-3q)³ = (12pq)³ - 3(12pq)(3q)(12pq-3q) - (3q)³

(12pq-3q)³ = (12pq)³ - 108pq²(12pq-3q) - (3q)³

(12pq-3q)³ = 1728p³q³ - 1296pq³ + 324pq³ - 27q³

Some more identities :-

(a+b)³ = a³ + 3a²b + 3ab² + b³

a³+b³ = (a+b)(a²-ab+b²)

a³-b³ = (a-b)(a²+ab+b²)

a³+b³+c³-3abc = (a+b+c)(a²+b²+c²-ab-bc-ca)

Conditional identity :-

if a+b+c = 0,

a³+b³+c³ = 3abc