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Find the remainder when f(x) = x⁵ - x³ + 3x² + 3x + 1 is divided by (x² - 1).

1. 3x + 4
2. x - 1
3. 2x + 1
4. 2x - 1

Correct answer: 3x + 4

Solution

Divisor x² - 1 = (x-1)(x+1), so remainder = p x + q. f(1) = 1 - 1 + 3 + 3 + 1 = 7 = p + q. f(-1) = -1 + 1 + 3 - 3 + 1 = 1 = -p + q. Adding: 2q = 8 => q = 4; subtracting: 2p = 6 => p = 3. Remainder = 3x + 4.

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