(x-2)^3-6(x+4)(x-4)-(x-2)(x^2+2x+4)

2 min read Jun 17, 2024
(x-2)^3-6(x+4)(x-4)-(x-2)(x^2+2x+4)

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.

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