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Cube of difference

Definition.
The сube of difference of two expressions is equal to the сube of the first, minus three times the product of the square of the first and the second, plus three times the product of the first and the square of second, minus the сube of the second:

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

Derivation of the formula of cube of difference

The proof of the formula is very simple. To prove the formula is sufficient to multiply the expression:

(a - b)3 = (a - b)·(a - b)2 =

= (a - b)·(a2 - 2ab + b2) =

= a3 - 2a2b + ab2 - ba2 + 2b2a - b3 =

= a3 - 3a2b + 3ab2 - b3

Applying of cube of difference formula

Cube of difference formula convenient to use:
• to disclose the brackets
• to simplify expressions

Example 1.
Expand brackets (x - 3)3.

Solution: Apply the cube of difference formula.

(x - 3)3 = x3 - 3·3·x2 + 3·32·x - 33 =

= x3 - 9x2 + 27x - 27
Example 2.
Expand brackets (2x - 3y2)3.

Solution: Apply the cube of difference formula.

(2x - 3y2)3 =

= (2x)3 - 3·(2x)2·(3y2) + 3·(2x)·(3y2)2 - (3y2)3 =

= 8x3 - 36x2y2 + 54xy4 - 27y6
Example 3.
Simplify the expression 27x3 - 27x2 + 9x - 19x2 - 6x + 1.

Solution: Apply the cube of difference formula in numerator and square of the difference formula in denominator.

27x3 - 27x2 + 9x - 19x2 - 6x + 1 = (3x - 1)3(3x - 1)2 = 3x - 1