A MINUS B WHOLE CUBE FORMULA

In this section, we are going to see the formula/expansion for

(a - b)³

That is,

(a - b)³ = (a - b)(a - b)(a - b)

Multiply (a - b) and (a - b).

(a - b)³ = (a² - ab - ab + b²)(a - b)

Simplify.

(a - b)³ = (a² - 2ab + b²)(a - b)

(a - b)³ = a³ - a²b - 2a²b + 2ab² + ab² - b³

Combine the like terms.

(a - b)³ = a³ - 3a²b + 3ab² - b³

or

(a - b)³ = a³ - b³ - 3ab(a - b)  

Solved Problems

Problem 1 :

Expand :

(x - 1)³

Solution :

(x - 1)³ is in the form of (a - b)³

Comparing (a - b)³ and (x - 1)³, we get

a = x

b = 1

Write the formula / expansion for (a - b)³.

(a - b)³ = a³ - 3a²b + 3ab² - b³

Substitute x for a and 1 for b.

(x - 1)³ = x³ - 3(x²)(1) + 3(x)(1²) - 1³

(x - 1)³ = x³ - 3x² + 3(x)(1) - 1

(x - 1)³ = x³ - 3x² + 3x - 1

So, the expansion of (x - 1)³ is

x³ - 3x² + 3x - 1

Problem 2 :

Expand :

(2x - 3)³

Solution :

(2x - 3)³ is in the form of (a - b)³

Comparing (a - b)³ and (2x - 3)³, we get

a = 2x

b = 3

Write the formula / expansion for (a - b)³.

(a - b)³ = a³ - 3a²b + 3ab² - b³

Substitute 2x for a and 3 for b.

(2x - 3)³ = (2x)³ - 3(2x)²(3) + 3(2x)(3²) - 3³

(2x - 3)³ = 8x³ - 3(4x²)(3) + 3(2x)(9) - 27

(2x - 3)³ = 8x³ - 36x² + 54x - 27

So, the expansion of (2x - 3)³ is

8x³ - 36x² + 54x - 27

Problem 3 :

Expand :

(x - 2y)³

Solution :

(x - 2y)³ is in the form of (a - b)³

Comparing (a - b)³ and (x - 2y)³, we get

a = x

b = 2y

Write the formula / expansion for (a - b)³.

(a - b)³ = a³ - 3a²b + 3ab² - b³

Substitute x for a and 2y for b.

(x - 2y)³ = x³ - 3(x²)(2y) + 3(x)(2y)² - (2y)³

(x - 2y)³ = x³ - 6x²y + 3(x)(4y²) - 8y³

(x - 2y)³ = x³ - 6x²y + 12xy² + 8y³

So, the expansion of (x - 2y)³ is

x³ - 6x²y + 12xy² + 8y³

Problem 4 :

If a - b = 3 and a³ - b³ = 1197, then find the value of ab.

Solution :

To find the value of ab, we can use the formula or expansion for (a - b)³.

Write the formula / expansion for (a - b)³.

(a - b)³ = a³ - 3a²b + 3ab² - b³

or

(a - b)³ = a³ - b³ - 3ab(a - b)

Substitute 13 for (a - b) and 1197 for (a³ - b³).

(3)³ = 1197 - 3(ab)(13)

Simplify.

27 = 1197 - 39ab

Subtract 1197 from each side.

-1170 = -39ab

Divide each side by (-39).

30 = ab

So, the value of ab is 30.

Problem 5 :

Find the value of :

(98)³

Solution :

We can use the algebraic formula for (a - b)³ and find the value of (98)³ easily.

Write (98)³ in the form of (a - b)³.

(98)³ = (100 - 2)³

Write the expansion for (a - b)³.

(a - b)³ = a³ - b³ - 3ab(a - b)

Substitute 100 for a and 2 for b.

(100 - 2)³ = 100³ - 2³ - 3(100)(2)(100 - 2)

(100 - 2)³ = 1000000 - 8 - 3(100)(2)(98)

(100 - 2)³ = 1000000 - 8 - 58800

(98)³ = 941192

So, the value of (107)³ is

941,192

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