- We perform polynomial long division on the dividend \((2x^4 - 3x^3 - 3x^2 + 7x - 3)\) and the divisor \((x^2 - 2x + 1)\).
- We divide the leading terms successively to find each term of the quotient.
- The division confirms the quotient is \(2x^2 + x - 3\) with a remainder of \(0\).
- The final answer is \(\boxed{2x^2 + x - 3}\).
Explanation
- Problem Analysis
To determine the quotient of the polynomial division \((2x^4 - 3x^3 - 3x^2 + 7x - 3) \div (x^2 - 2x + 1)\), we use the method of polynomial long division. Let the dividend be \(P(x) = 2x^4 - 3x^3 - 3x^2 + 7x - 3\) and the divisor be \(D(x) = x^2 - 2x + 1\). The quotient \(Q(x)\) is the polynomial such that \(P(x) = Q(x)D(x) + R(x)\), where the degree of the remainder \(R(x)\) is less than the degree of the divisor.
- Performing Polynomial Long Division
- Divide the leading term of the dividend \(2x^4\) by the leading term of the divisor \(x^2\): \(\frac{2x^4}{x^2} = 2x^2\). This is the first term of our quotient.
- Multiply this term by the divisor: \(2x^2(x^2 - 2x + 1) = 2x^4 - 4x^3 + 2x^2\).
- Subtract this from the original dividend: \((2x^4 - 3x^3 - 3x^2 + 7x - 3) - (2x^4 - 4x^3 + 2x^2) = x^3 - 5x^2 + 7x - 3\).
- Divide the leading term of the new expression \(x^3\) by the leading term of the divisor \(x^2\): \(\frac{x^3}{x^2} = x\). This is the second term of our quotient.
- Multiply this term by the divisor: \(x(x^2 - 2x + 1) = x^3 - 2x^2 + x\).
- Subtract this from the current expression: \((x^3 - 5x^2 + 7x - 3) - (x^3 - 2x^2 + x) = -3x^2 + 6x - 3\).
- Divide the leading term of the new expression \(-3x^2\) by the leading term of the divisor \(x^2\): \(\frac{-3x^2}{x^2} = -3\). This is the third term of our quotient.
- Multiply this term by the divisor: \(-3(x^2 - 2x + 1) = -3x^2 + 6x - 3\).
- Subtract this from the current expression: \((-3x^2 + 6x - 3) - (-3x^2 + 6x - 3) = 0\). The remainder is 0.
- Conclusion and Final Result
By collecting the terms calculated during the division, the quotient is \(2x^2 + x - 3\). Comparing this result with the given choices:
- \(2x^4 - 3x^3 - \frac{11}{2}\)
- \(2x^2 + x - 3 - \frac{6}{x^2 - 2x + 1}\)
- \(2x^2 - x\)
- \(2x^2 + x - 3\)
The correct option is \(2x^2 + x - 3\). Therefore, the final answer is \(\boxed{2x^2 + x - 3}\).
Examples
Polynomial division is a powerful tool in algebra, used frequently when analyzing the behavior of rational functions or simplifying complex expressions. For example, if you are designing a computer graphics algorithm to track the path of an object, you might need to divide complex trajectories represented by polynomials to simplify calculations. By mastering this technique, you can efficiently break down high-degree polynomials into manageable parts, much like the steps taken here to find the quotient \(2x^2 + x - 3\).