Polynomial Division by product of two polynomials

Question

f(x) is a polynomial, when it is divided by (x-3) it leaves remainder 15. when f(x) is divided by square of(x-1) it leaves remainder 2x+1. Find the remainder when f(x) is divided by product of two above divisors.

Answer 1

HINT:

Let $f(x)=g(x)(x-3)(x-1)^2+Ax^2+Bx+C$

$15=f(3)=0+3^2A+3B+C$

$2x+1=A\{x^2-(x-1)^2\}+Bx+C$

$\iff2x+1=x(B+2A)+C-A$

Compare the constants & the coefficients of $x$