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)is in the form of (a - b)3

Comparing (a - b)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)is in the form of (a - b)3

Comparing (a - b)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)is in the form of (a - b)3

Comparing (a - b)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)and find the value of (98)easily.

Write (98)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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