%5E3-(x%2B2y)(x%5E2-2xy%2B4y%5E2)%2B6x%5E2y.jpeg "(x-2y)^3-(x+2y)(x^2-2xy+4y^2)+6x^2y")
Expanding and Simplifying the Expression: (x-2y)^3 - (x+2y)(x^2-2xy+4y^2) + 6x^2y
This article will guide you through the process of simplifying the given expression:
(x-2y)^3 - (x+2y)(x^2-2xy+4y^2) + 6x^2y
We'll use the following algebraic rules:
- Binomial Theorem: (a + b)³ = a³ + 3a²b + 3ab² + b³
- Difference of Squares: a² - b² = (a + b)(a - b)
- Distributive Property: a(b + c) = ab + ac
Step 1: Expand (x-2y)³
Using the binomial theorem, we get:
(x - 2y)³ = x³ + 3(x²)(-2y) + 3(x)(-2y)² + (-2y)³ = x³ - 6x²y + 12xy² - 8y³
Step 2: Expand (x+2y)(x²-2xy+4y²)
Notice that this is a special case of the difference of cubes: a³ - b³ = (a - b)(a² + ab + b²)
In this case, a = x and b = 2y. Therefore:
(x + 2y)(x² - 2xy + 4y²) = x³ + (2y)³ = x³ + 8y³
Step 3: Substitute the expanded terms into the original expression
The original expression now becomes:
(x³ - 6x²y + 12xy² - 8y³) - (x³ + 8y³) + 6x²y
Step 4: Simplify the expression
Combining like terms, we get:
x³ - 6x²y + 12xy² - 8y³ - x³ - 8y³ + 6x²y = -6x²y + 12xy² - 16y³
Final Result:
The simplified form of the given expression is -6x²y + 12xy² - 16y³.