<s>[INST] Polynomials factor Theorem If a cubic polynomial $p(x)$ with leading co-efficient 1 is divided by $x-1$, $x-2$ and $x-3$ respectively it leaves $r(x)$ $1$, $4$ and $9$. What will be the remainder when it is divided by $x-4$?
I've searched for this everywhere but I couldn't find an answer. 
Help would be much appreciated. 
 [/INST] Let $f(x)=p(x)-x^2$. Then $f(x)$ is a cubic polynomial.
Note that $f(1)=p(1)-1^2=1-1=0$, $f(2)=p(2)-2^2=4-4=0$ and $f(3)=p(3)-3^2=9-9=0$
$f(x)$ has roots $1$, $2$ and $3$. So,
$$f(x)=A(x-1)(x-2)(x-3)$$
for some constant $A$.
$$p(x)=A(x-1)(x-2)(x-3)+x^2$$
When $p(x)$ is divided by $x-4$, then remainder is
$$p(4)=6A+16$$
Without other information on $p(x)$, we cannot proceed further.
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