(x%5E2%2B2xy%2B4y%5E2)%2B(x%2B2y)(x%5E2-2xy%2B4y%5E2).jpeg "(x-2y)(x^2+2xy+4y^2)+(x+2y)(x^2-2xy+4y^2)")
Simplifying the Expression: (x-2y)(x^2+2xy+4y^2)+(x+2y)(x^2-2xy+4y^2)
This expression presents a neat opportunity to use the difference of squares pattern and the sum of cubes pattern to simplify it. Let's break it down step-by-step:
Recognizing the Patterns
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Difference of Squares: The expression (x^2 + 2xy + 4y^2) resembles the pattern (a^2 + ab + b^2) which is a part of the difference of cubes pattern. Similarly, (x^2 - 2xy + 4y^2) also fits the same pattern.
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Sum of Cubes: The expression (x - 2y)(x^2 + 2xy + 4y^2) represents the factorization of (x^3 - (2y)^3). Likewise, (x + 2y)(x^2 - 2xy + 4y^2) represents the factorization of (x^3 + (2y)^3).
Applying the Patterns
Using the above observations, we can rewrite the expression as follows:
(x-2y)(x^2+2xy+4y^2)+(x+2y)(x^2-2xy+4y^2) = (x^3 - (2y)^3) + (x^3 + (2y)^3)
Simplifying Further
Now, we can simplify the expression by combining the terms:
(x^3 - (2y)^3) + (x^3 + (2y)^3) = 2x^3
Therefore, the simplified form of the given expression is 2x^3.