SOLUTION: 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-

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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