Question

divide 3x2-x3 -3x+5 by x-1-x2 &verify division algorithm

Answer 1

Negate and rearrange dividend and divisor: x³-3x²+3x-5 and x²-x+1.

Division:

              x  -2             

x²-x+1 ) x³-3x²+3x-5

              x³ -x²  +x

                 -2x²+2x-5

                 -2x²+2x-2

                             -3 remainder. This remainder must also be negated, so remainder = 3. The reason is that we negated the divisor and the remainder is in relation to the divisor: -3/(x²-x+1)=3/(x-1-x²).

CHECK

(x-2)(x-1-x²)+3=x²-x-x³-2x+2+2x²+3=3x²-x³-3x+5.