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Sum of cubes

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

a3 + b3 = (a + b)·(a2 - ab + b2)


Derivation of the formula of sum 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 sum of cubes formula

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

Examples of task

Example 1.
Factorised x3 + 27.

Solution: Apply the sum of cubes formula.

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

Solution: Apply the sum 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 sum of cubes formula in numerator.

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

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