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

3 min read Jun 16, 2024
(-x-2)^3+(2x-4)(x^2+2x+4)-x^2(x-6)

Simplifying the Expression (-x-2)^3 + (2x-4)(x^2+2x+4) - x^2(x-6)

This article aims to guide you through the process of simplifying the given algebraic expression: (-x-2)^3 + (2x-4)(x^2+2x+4) - x^2(x-6). We will break down the problem into manageable steps and use fundamental algebraic rules.

Step 1: Expanding the Cubes and Products

  • Expanding (-x-2)^3:

    • Recall the cube of a binomial formula: (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
    • Applying this to our expression, we get: (-x-2)^3 = (-x)^3 + 3(-x)^2(-2) + 3(-x)(-2)^2 + (-2)^3 = -x^3 + 6x^2 - 12x - 8

  • Expanding (2x-4)(x^2+2x+4):

    • This is a product of two binomials. We can expand it using the distributive property (FOIL method) or by recognizing it as a sum of cubes pattern.
    • Let's use the sum of cubes pattern: (a-b)(a^2 + ab + b^2) = a^3 - b^3
    • Applying this, we get: (2x-4)(x^2+2x+4) = (2x)^3 - (4)^3 = 8x^3 - 64

  • Expanding -x^2(x-6):

    • Simple distribution: -x^2(x-6) = -x^3 + 6x^2

Step 2: Combining Like Terms

Now, we have the expression: -x^3 + 6x^2 - 12x - 8 + 8x^3 - 64 - x^3 + 6x^2

Combine the like terms:

  • x^3 terms: -x^3 + 8x^3 - x^3 = 6x^3
  • x^2 terms: 6x^2 + 6x^2 = 12x^2
  • x terms: -12x
  • Constant terms: -8 - 64 = -72

Step 3: The Simplified Expression

Finally, combining all the simplified terms, we get the simplified expression:

6x^3 + 12x^2 - 12x - 72

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