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Cube of sum

Definition.
The сube of sum of two expressions is equal to the сube of the first, plus 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, plus the сube of the second:

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


Derivation of the formula of cube of sum

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 sum formula

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

Examples of task

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

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

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

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