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Solved Examples: Cube of a Difference `(a - b)^3`

Solved Example 1

Expand `(x -2y)^3`

Solution

Step 1: Compare the given expression with `(a - b)^3`

Comparing  `(x - 2y)^3` with `(a - b)^3`, we get
`a = x` and `b = 2y`

Step 2: Apply the formula `(a - b)^3 = a^3 - b^3 - 3ab(a - b)`

Putting `x` in place of `a` and `2y` in place of `b` in the above formula, we get,
`(x - 2y)^3 = x^3 - (2y)^3 - 3(x)(2y)(x - 2y)`
Simplifying the left hand expression,
`(x - 2y)^3 = x^3 - 8y^3 - 6xy(x - 2y)`
`= x^3 - 8y^3 - 6x^2y + 12xy^2`

Solved Example 2

Expand `(3x -2y)^3`

Solution

Step 1: Compare the given expression with `(a - b)^3`

Comparing  `(3x - 2y)^3` with `(a - b)^3`, we get
`a = 3x` and `b = 2y`

Step 2: Apply the formula `(a - b)^3 = a^3 - b^3 - 3ab(a - b)`

Putting `3x` in place of `a` and `2y` in place of `b` in the above formula, we get,
`(3x - 2y)^3 = (3x)^3 - (2y)^3 - 3(3x)(2y)(3x - 2y)`
Simplifying the left hand expression,
`(3x - 2y)^3 = 27x^3 - 8y^3 - 18xy(3x - 2y)`
`= 27x^3 - 8y^3 - 54x^2y + 36xy^2`

Solved Example 3

Expand `(x/3 - 2y)^3`

Solution

Step 1: Compare the given expression with `(a - b)^3`

Comparing  `(x/3 - 2y)^3` with `(a - b)^3`, we get
`a = x/3` and `b = 2y`

Step 2: Apply the formula `(a - b)^3 = a^3 - b^3 - 3ab(a - b)`

Putting `x/3` in place of `a` and `2y` in place of `b` in the above formula, we get,
`(x/3 - 2y)^3 = (x/3)^3 - (2y)^3 - 3(x/3)(2y)(x/3 - 2y)`
Simplifying the left hand expression,
`(x/3 - 2y)^3 = (x^3)/27 - 8y^3 - 2xy(x - 2y)`
`= (x^3)/27 - 8y^3 - 2x^2y + 4xy^2`

Solved Example 4

Expand `(a/2 - b/3)^3`

Solution

Step 1: Compare the given expression with `(a - b)^3`

Comparing  `(a/2 - b/3)^3` with `(a - b)^3`, we get
`a = a/2` and `b = b/3`

Step 2: Apply the formula `(a - b)^3 = a^3 - b^3 - 3ab(a - b)`

Putting `a/2` in place of `a` and `b/3` in place of `b` in the above formula, we get,
`(a/2 - b/3)^3 = (a/2)^3 - (b/3)^3 - 3(a/2)(b/3)(a/2 - b/3)`
Simplifying the left hand expression,
`(a/2 - b/3)^3 = (a^3)/8 - (b^3)/27 - (ab)/2(a/2 - b/3)`
`= (a^3)/8 - (b^3)/27 - (a^2b)/4 + (ab^2)/6`

Solved Example 5

Expand `(x - 1)^3`

Solution

Step 1: Compare the given expression with `(a - b)^3`

Comparing  `(x - 1)^3` with `(a - b)^3`, we get
`a = x` and `b = 1`

Step 2: Apply the formula `(a - b)^3 = a^3 - b^3 - 3ab(a - b)`

Putting `x` in place of `a` and `1` in place of `b` in the above formula, we get,
`(x - 1)^3 = x^3 - 1^3 - 3(x)(1)(x - 1)`
Simplifying the left hand expression,
`(x - 1)^3 = x^3 - 1 - 3x(x - 1)`
`= x^3 - 1 - 3x^2 + 3x`

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