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Simplifying the Expression (x-2)³ - 6(x+4)(x-4) - (x-2)(x²+2x+4)
This article will guide you through simplifying the algebraic expression:
(x-2)³ - 6(x+4)(x-4) - (x-2)(x²+2x+4)
We will achieve this simplification by applying the following:
- Expanding the cubes: We will use the formula (a-b)³ = a³ - 3a²b + 3ab² - b³
- Difference of Squares: We will use the formula (a+b)(a-b) = a² - b²
- Sum of Cubes: We will use the formula (a-b)(a²+ab+b²) = a³ - b³
Let's start by expanding each term:
1. (x-2)³:
Applying the formula, we get:
(x-2)³ = x³ - 3x²(2) + 3x(2)² - 2³ = x³ - 6x² + 12x - 8
2. 6(x+4)(x-4):
Using the difference of squares formula, we obtain:
6(x+4)(x-4) = 6(x² - 4²) = 6(x² - 16) = 6x² - 96
3. (x-2)(x²+2x+4):
Here, we can use the sum of cubes formula:
(x-2)(x²+2x+4) = x³ - 2³ = x³ - 8
Now, let's substitute these simplified terms back into the original expression:
(x³ - 6x² + 12x - 8) - (6x² - 96) - (x³ - 8)
Finally, we can combine like terms:
x³ - 6x² + 12x - 8 - 6x² + 96 - x³ + 8
= -12x² + 12x + 96
Therefore, the simplified form of the expression (x-2)³ - 6(x+4)(x-4) - (x-2)(x²+2x+4) is -12x² + 12x + 96.