Final answer:
The division of the polynomial (2x⁴ - 3x³ - 3x² + 7x - 3) by (x² - 2x + 1) through polynomial long division results in the quotient of 2x² - x - 3, which matches option B).
Step-by-step explanation:
To divide the polynomial (2x⁴ - 3x³ - 3x² + 7x - 3) by the polynomial (x² - 2x + 1), we use polynomial long division.
First, we divide the first term of the numerator by the first term of the denominator: 2x⁴ ÷ x² = 2x². We then multiply 2x² by (x² - 2x + 1) and subtract the result from the original polynomial.
We continue this process with the new polynomial generated after subtraction, taking its first term and dividing again by the first term of the denominator, multiplying the result by the whole denominator and subtracting again until we have a degree in the resultant polynomial that is less than the degree of the denominator or there is no remainder.
The final answer is 2x² - x - 3, which corresponds to option B).