SOLUTION: When f(x)is divided by x-1, the remainder is -1; when it is divided by {{{x^2}}}, the remainder is -x-1. Find the remainder when f(x)is divided by {{{(x^2)(x-1)}}}

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: When f(x)is divided by x-1, the remainder is -1; when it is divided by {{{x^2}}}, the remainder is -x-1. Find the remainder when f(x)is divided by {{{(x^2)(x-1)}}}      Log On


   

Question 1036408: When f(x)is divided by x-1, the remainder is -1; when it is divided by x%5E2, the remainder is -x-1. Find the remainder when f(x)is divided by %28x%5E2%29%28x-1%29
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
The remainder would be x%5E2+-+x+-+1.
To see this, notice that the polynomial f%28x%29+=+x%5E%282n-1%29+-+x+-+1, where n%3E=2, will always give a remainder of -1 upon division by x-1 and a remainder of -x-1 upon division by x%5E2. By using synthetic division with x - 1 as divisor, it can be easily seen that f(x) is unique in form.
By applying the usual polynomial division, the remainder after dividing f%28x%29+=+x%5E%282n-1%29+-+x+-+1 by x%5E2%28x-1%29+=+x%5E3+-+x%5E2 is x%5E2+-+x+-+1.