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

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

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

This article will walk you through simplifying the given algebraic expression.

Understanding the Expression:

The expression contains a combination of:

  • Cubic terms: (x-1)^3
  • Quadratic terms: 4x(x+1)(x-1)
  • Linear terms: 3(x-1)(x^2+x+1)

Each term involves multiplications and powers of the variable 'x'. Our goal is to simplify this expression by expanding and combining like terms.

Step-by-Step Simplification:

  1. Expand the cubic term:

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

  2. Expand the quadratic term:

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

  3. Expand the linear term:

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

  4. Combine all the terms:

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

  5. Simplify by combining like terms:

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

Final Result:

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

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