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Definition. Difference of cubes of two expressions can be found using the following formula:
a^{3}  b^{3} = (a  b)·(a^{2} + ab + b^{2}) Derivation of the formula of difference of cubesThe proof of the formula is very simple. To prove the formula is sufficient to multiply the expression: (a  b)·(a^{2} + ab + b^{2}) =
= a^{3} + a^{2}b + ab^{2}  ba^{2}  ab^{2}  b^{3} = a^{3}  b^{3} Applying of difference of cubes formula
Difference of cubes formula convenient to use:
Examples of taskExample 1. Factorised x^{3}  27.
Solution: Apply the difference of cubes formula. x^{3}  27 = x^{3}  3^{3} = (x  3)·(x^{2} + 3x + 9)
Example 2. Factorised 8x^{3}  27y^{6}.
Solution: Apply the difference of cubes formula. 8x^{3}  27y^{6} = (2x)^{3}  (3y^{2})^{3} =
= (2x  3y^{2})·(4x^{2} + 6xy^{2} + 9y^{4}) Example 3. Simplify the expression Solution: Apply the difference 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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