Verified Answer :

  • Divide the leading term \(x^3\) by \(x\) to get \(x^2\), subtract \(x^2(x-1) = x^3-x^2\) from the dividend to get \(-2x+1\).
  • Divide the new leading term \(-2x\) by \(x\) to get \(-2\), subtract \(-2(x-1) = -2x+2\) to get the final remainder \(-1\).
  • The quotient is \(q(x) = x^2 - 2\) and the remainder is \(r = -1\).
  • Final expression: \(\boxed{q(x) = x^2 - 2, r = -1}\).

Explanation

  1. Problem Analysis

The objective is to perform polynomial long division on the expression \(\frac{x^3 - x^2 - 2x + 1}{x - 1}\). We identify the dividend as \(P(x) = x^3 - x^2 - 2x + 1\) and the divisor as \(D(x) = x - 1\). Polynomial long division will allow us to write the expression in the form \(Q(x) + \frac{R}{x - 1}\), where \(Q(x)\) is the quotient polynomial and \(R\) is the remainder constant.

  1. Performing Polynomial Long Division

We start the division process by focusing on the leading terms:

  1. Divide the leading term of the dividend, \(x^3\), by the leading term of the divisor, \(x\): \(\frac{x^3}{x} = x^2\). This is the first term of our quotient.
  2. Multiply this term by the divisor: \(x^2(x - 1) = x^3 - x^2\).
  3. Subtract this from the original dividend: \((x^3 - x^2 - 2x + 1) - (x^3 - x^2) = -2x + 1\).
  4. Now, divide the leading term of the remainder, \(-2x\), by the leading term of the divisor, \(x\): \(\frac{-2x}{x} = -2\). This is the second term of our quotient.
  5. Multiply this term by the divisor: \(-2(x - 1) = -2x + 2\).
  6. Subtract this from the current remainder: \((-2x + 1) - (-2x + 2) = -1\).

Since the degree of \(-1\) (which is 0) is less than the degree of \(x - 1\) (which is 1), the division process is complete. The remainder is \(-1\).

  1. Conclusion

Based on our division steps, the quotient \(Q(x)\) is \(x^2 - 2\) and the remainder \(R\) is \(-1\). Thus, we can express the original fraction as:
\[\frac{x^3 - x^2 - 2x + 1}{x - 1} = x^2 - 2 - \frac{1}{x - 1}\]
The quotient is \(q(x) = x^2 - 2\) and the remainder is \(r = -1\).

Examples

Polynomial division is a fundamental technique used in calculus and algebra for simplifying rational functions, which are ratios of two polynomials. For example, when you need to integrate a rational function like \(\int \frac{x^3 - x^2 - 2x + 1}{x - 1} dx\), performing long division first simplifies the integrand to \(\int (x^2 - 2 - \frac{1}{x-1}) dx\), which is much easier to compute. This skill is essential for analyzing function behavior, graphing curves, and solving complex engineering or physical problems where variables relate through polynomials.

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