Question

Remainder of x⁵-2x⁴ x³ 3x²-2x 1 when divide by x²-1
A.0
B.2x-1
C.x-2
D.x²+1

Answer 1

To find the remainder when dividing x⁵-2x⁴+x³-3x²-2x+1 by x²-1, use polynomial long division to divide the terms. The remainder is -x+6.

Step-by-step explanation:

To find the remainder when dividing x⁵-2x⁴+x³-3x²-2x+1 by x²-1, we can use polynomial long division.

1. Divide the leading terms: x³ ÷ x² which gives x.
2. Multiply the quotient obtained by the entire divisor: x · (x²-1) = x³-x.
3. Subtract the result of the previous step from the original polynomial: x³-2x⁴+x³-3x²-2x+1 - (x³-x) = -2x⁴-2x³-3x²-x+1.
4. Repeat the process with the new polynomial obtained, until no more terms can be divided.

Continuing with the division, we get:

• -2x⁴ ÷ x² = -2x²
• -2x² · (x²-1) = -2x⁴+2x²
• -2x⁴-2x³-3x²-x+1 - (-2x⁴+2x²) = -2x³-5x²-x+1
• -2x³ ÷ x² = -2x
• -2x · (x²-1) = -2x³+2x
• -2x³-5x²-x+1 - (-2x³+2x) = -5x²-x+1
• -5x² ÷ x² = -5
• -5 · (x²-1) = -5x²+5
• -5x²-x+1 - (-5x²+5) = -x+6

Since the resulting polynomial -x+6 has a degree less than the divisor's degree, the remainder is -x+