Final answer:
The quotient of (x⁴ - 3x³ - 3x - 1) divided by (x² - 1) through polynomial long division is x² - 3x + 3, with a remainder of -4x + 2.
Step-by-step explanation:
To find the quotient of (x⁴ - 3x³ - 3x - 1) divided by (x² - 1), we perform polynomial long division or synthetic division. Since the divisor is a quadratic, polynomial long division is the more suitable approach. The division process is similar to long division of numbers. We determine how many times the first term of the divisor x² fits into the first term of the dividend x⁴, which is x² times. Multiplying the divisor by x² and subtracting it from the dividend, we proceed to the next term.
The complete polynomial long division process will give us the quotient:
x² - 3x + 3 will be the quotient and -4x + 2 as the remainder. So the final answer is x² - 3x + 3 + (-4x + 2)/(x² - 1).