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

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

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

This article aims to simplify the given expression by expanding and combining like terms.

Step 1: Expanding the Expressions

We begin by expanding both products in the expression:

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

  • Multiply each term in the first set of parentheses by each term in the second set:
    • x^3 * (x-1) = x^4 - x^3
    • -2x^2 * (x-1) = -2x^3 + 2x^2
    • 3x * (x-1) = 3x^2 - 3x
    • -4 * (x-1) = -4x + 4

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

  • Multiply each term in the first set of parentheses by each term in the second set:
    • 2x * (x^2-x+1) = 2x^3 - 2x^2 + 2x
    • -3 * (x^2-x+1) = -3x^2 + 3x - 3

Step 2: Combining Like Terms

Now we have:

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

Combine like terms:

  • x^4 + (-1-2)x^3 + (2+3)x^2 + (-3-4)x + 4 - (2x^3 - 5x^2 + 5x -3)

Simplify:

  • x^4 - 3x^3 + 5x^2 - 7x + 4 - 2x^3 + 5x^2 - 5x + 3

Step 3: Final Simplification

Combine the like terms again:

  • x^4 - 5x^3 + 10x^2 - 12x + 7

Therefore, the simplified form of the expression (x^3-2x^2+3x-4)(x-1)-(2x-3)(x^2-x+1) is x^4 - 5x^3 + 10x^2 - 12x + 7.

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