You can factorize expressions of form a3 + b3 and a3 - b3 into simpler forms using the algebraic identities sum and difference of cubes. When simplifying algebraic expressions and resolving equations, these identities are helpful.
These identities can be obtained by extending the expressions on the right and comparing them with the expressions on the left.
As an example, to factorize the expression x^3 + 8^3, the sum of cubes identity can be used:
x^3 + 8^3 = (x + 8)(x^2 - 8x + 8^2) = (x + 8)(x^2 - 8x + 64)
To simplify algebraic expressions or to solve equations you can use these identities. As an example, to find the value of x that satisfies the equation x^3 + 8^3 = 0, the sum of cubes identity can be used to write:
x^3 + 8^3 = (x + 8)(x^2 - 8x + 64) = 0
This equation can be expressed as (x + 8) = 0 or (x^2 - 8x + 64) = 0. Once you solve these equations singly it will give you x = -8 and x = 4, which are the original equation solutions.
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