<s>[INST] Remainder of the polynomial A polynomial function $f(x)$  with real coefficients leaves the remainder $15$ when divided by $x-3$, and the remainder $2x+1$ when divided by $(x-1)^2$. Then the remainder when $f(x)$ is divided by $(x-3)(x-1)^2$ is?
What I have thought-The remainder must be of the form $ax^2+bx+c$. Now applying the remainder theorem, I am able to find $2$ equations in $a,b,c$ . eg
Let $f(x)$=$(x-3)$$h(x)$+$15$             ...................................(1)
Also let $f(x)=(x-3)(x-1)^2g(x)+ax^2+bx+c$...........................(2)
Put $x=3$ and using (1) we get $15=9a+3b+c$ 
Similarly I can get another equation using the other information given. But I am only able to get 2 equations in 3 variables. From where do I get the 3rd equation in $a,b,c$ and hence the remainder?
 [/INST] Hint $\ f(x) = 2x\!+\!1+ (x\!-\!1)^2 (c + (x-3)g(x))\,$ and $\,15 = f(3) = 7+4c\ $ so $\ c = \,\ldots$
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