Sum of cubes
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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
= 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)
= (2x + 3y2)·(4x2 - 6xy2 + 9y4)
Example 3.
Simplify the expression Solution: Apply the sum of cubes formula in numerator.
Factoring: Some special cases
Square of the sum
Square of the difference
Difference of squares
Cube of sum
Cube of difference
Sum of cubes
Difference of cubes
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