Question 1049425: a polynomial f(x) with rational coefficients leaves remainder 15 when divided by x-3 and remainder 2x+1 when divided by (x-1)^2. Find the remainder when f(X) is divided by (x-3)(x-1)^2.
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! Since divisor  is a polynomial of degree 3, the remainder we look for is a polynomial whose degree is at most 2.
Let us say that remainder is  , and we have to find coefficients  ,  , and  .
So, 
Since the remainder is  when divided by  ,  , so
When  is divided by  the remainder is the remainder of dividing by  .
That remainder is  .
Since the polynomials  and  are the same polynomial,
 <--->  .
Along with the equation  highlighted above, we have the system of linear equations
 .
Substituting into the top equation the expressions for and from the bottom two equations, we get
 , and substituting  for in the bottom two equations of the system, we get
   , and
  .
So, it turns out that the remainder that we were looking for is
 .
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