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(x-2)(x^2+2x+4)-(x-1)^3+7

2 min read Jun 17, 2024
(x-2)(x^2+2x+4)-(x-1)^3+7

Simplifying the Expression: (x-2)(x^2+2x+4)-(x-1)^3+7

This article will guide you through the process of simplifying the algebraic expression: (x-2)(x^2+2x+4)-(x-1)^3+7.

Understanding the Expression

The expression consists of three main components:

  1. (x-2)(x^2+2x+4): This is a product of two expressions. The second expression is a perfect square trinomial.
  2. -(x-1)^3: This represents the negative of a perfect cube.
  3. +7: This is a constant term.

Simplifying the Expression

Step 1: Expand the first term using the distributive property or by recognizing it as the difference of cubes pattern:

  • (x-2)(x^2+2x+4) = x^3 - 2^3 = x^3 - 8

Step 2: Expand the second term using the cube of a binomial formula:

  • -(x-1)^3 = -(x^3 - 3x^2 + 3x - 1)

Step 3: Distribute the negative sign:

  • -(x^3 - 3x^2 + 3x - 1) = -x^3 + 3x^2 - 3x + 1

Step 4: Combine all the terms:

  • x^3 - 8 - x^3 + 3x^2 - 3x + 1 + 7 = 3x^2 - 3x

Final Result

The simplified form of the expression (x-2)(x^2+2x+4)-(x-1)^3+7 is 3x^2 - 3x.

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