Q13/037) The remainder of f(x)/(x^2+x+1) and f(x)/[(x+1)^2] are x+5 and x-1 respectively.? What is the remainder of f(x)/[(x^2+x+1)(x+1)]

We have

f(x) = P(x)(x^2+x + 1)+ (x+5)  ..(1)
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and f(x) = Q(x)(x+1)^2 + (x-1) = (Q(x)(x+1) +1)(x +1) - 2 ..(2)

now f(x) divided by (x^2+x+1)(x+1) the remainder shall be a quadratic polynomial say

A(x^2 + x + 1) + Bx + C

from (1) B= 1 and C = 5

so remainder = A(x^2+x + 1) + x + 5
from (2) we should have A + 4 = - 2 or A = - 6

so remainder = - 6 x^2 - 5 x -1