Verified Answer :

  • Define the division: \((x^3 - 3x^2 + 3x - 2) \div (x^2 - x + 1)\).
  • Perform polynomial long division by identifying the quotient terms.
  • Determine the quotient \(Q(x) = x - 2\) with a remainder of 0.
  • Confirm the result matches the provided option \(\boxed{x - 2}\).

Explanation

  1. Analyzing the Polynomial Division Problem

We are tasked with finding the quotient of the polynomial division \((x^3 - 3x^2 + 3x - 2) \div (x^2 - x + 1)\). Polynomial division allows us to decompose a dividend \(P(x)\) by a divisor \(D(x)\) into the form \(P(x) = D(x) \cdot Q(x) + R(x)\), where \(Q(x)\) is the quotient and \(R(x)\) is the remainder. Our objective is to determine \(Q(x)\).

  1. Performing Polynomial Long Division

To divide \((x^3 - 3x^2 + 3x - 2)\) by \((x^2 - x + 1)\), we follow the standard long division process:

  1. Divide the leading term of the dividend (\(x^3\)) by the leading term of the divisor (\(x^2\)). This gives the first term of the quotient: \(x^3 / x^2 = x\).
  2. Multiply this term by the entire divisor: \(x(x^2 - x + 1) = x^3 - x^2 + x\).
  3. Subtract this result from the original dividend: \((x^3 - 3x^2 + 3x - 2) - (x^3 - x^2 + x) = -2x^2 + 2x - 2\).
  4. Now, divide the leading term of this new expression (\(-2x^2\)) by the leading term of the divisor (\(x^2\)). This gives the second term of the quotient: \(-2x^2 / x^2 = -2\).
  5. Multiply this term by the divisor: \(-2(x^2 - x + 1) = -2x^2 + 2x - 2\).
  6. Subtract this from the current remainder: \((-2x^2 + 2x - 2) - (-2x^2 + 2x - 2) = 0\).

Since the remainder is 0, the quotient is \(x - 2\).

  1. Conclusion and Final Answer

The division results in a quotient of \(x - 2\) with a remainder of 0. Comparing this result with the provided options, we find it matches the first option. The final answer is \(\boxed{x - 2}\).

Examples

Polynomial division is essential in many fields, such as engineering and physics, where complex systems are modeled using polynomial equations. For instance, when designing a control system, engineers often need to divide a transfer function's numerator by its denominator to analyze stability or determine the system's response to inputs. Understanding how to perform these calculations efficiently allows for the simplification of complex models, making it easier to solve problems and understand the underlying dynamics of various systems.

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