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Difference of cubes

Definition.
Difference of cubes of two expressions can be found using the following formula:

a3 - b3 = (a - b)·(a2 + ab + b2)


Derivation of the formula of difference of cubes

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

(a - b)·(a2 + ab + b2) =

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

Applying of difference of cubes formula

Difference of cubes formula convenient to use:
  • to factorised
  • to simplify expressions

Examples of task

Example 1.
Factorised x3 - 27.

Solution: Apply the difference of cubes formula.

x3 - 27 = x3 - 33 = (x - 3)·(x2 + 3x + 9)
Example 2.
Factorised 8x3 - 27y6.

Solution: Apply the difference of cubes formula.

8x3 - 27y6 = (2x)3 - (3y2)3 =

= (2x - 3y2)·(4x2 + 6xy2 + 9y4)
Example 3.
Simplify the expression 27x3 - 13x - 1.

Solution: Apply the difference of cubes formula in numerator.

27x3 - 13x - 1 = (3x - 1)·(9x2 + 3x +1)3x - 1 = 9x2 + 3x +1

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