<s>[INST] Find a polynomial given the remainders of the division of that polynomial with 3 other polynomials 
A polynomial from $ \mathbb{C}[x]$ divided by $ x - 1$, $x + 1$, $ x -2$ has the remainders 2, 6 and 3. Find the remainder of the division of that polynomial by $(x-1)(x+1)(x-2)$

The degree of the expression $(x-1)(x+1)(x-2)$ is 3, so the degree of the polynomial that I am looking for is 4. So the polynomial should be of the form: $ax^4 + bx^3 +cx^2 + dx + e$. I have tried to use Horner's scheme and I got the following expressions: $a+b+c+d+e=2$, $a+c-b-d+e=6$ and $16a+8b+4c+2d+c=3$. But this is not enough in order to find the polynomial. Is this the right approach?
 [/INST] Lets call this polynomial $P(x)$ than by the conditions you have that the polynomial $P(x)$ can be written as $$P(x)=(x-1)Q_1(x)+2\\P(x)=(x+1)Q_2(x)+6\\P(x)=(x-2)Q_3(x)+3$$ From this you can see that $P(1)=2,P(-1)=6,P(2)=3$
Now you can also write your polynomial as
$$P(x)=(x-1)(x+1)(x-2)Q_4(x)+ax^2+bx+c$$
Where $ax^2+bx+c$ is reminder when $P(x)$ is divided by $(x-1)(x+1)(x-2)$ plugging in $x=1,-1,2$ you can find $a,b,c$
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