Remainder theorem.

Since you raised this general question and I wanted to get a general procedure for this, I spent almost all of my evening thinking about this. This is what I came up with. Maybe this might help you.

Consider the polynomial $p(x)$ of degree $d_p$ and the divisor polynomial $d(x)$ of degree $d_q$ such that $d_q\leq d_p$ (obviously) where we have $d_p,d_q\in\Bbb{Z_0^+}$. Denote by $q(x)$ the quotient polynomial you get while performing division algorithm and denote by $r(x)$ the remainder you get when $p(x)$ is divided by $d(x)$.

Note: If $d_p=0$, then the polynomial is a constant function which is divisible by any divisor polynomial of degree $0$. Also, if we have $d_q=0$, we simply have that the divisor is a constant function which obviously divides polynomials $p(x)$ of any degree. If $d_p=d_q=0$, the result is the same which follows from the already discussed two cases. So, these trivial cases can be dismissed from our examination since they always give the remainder as $0$.

We proceed to get a systematic procedure to deal with the non-trivial cases, i.e., when $d_p,d_q\in\Bbb{Z^+}$.

Consider the set $\large\{r_i\}_{i=1}^{i=q_d}$ of all roots of $d(x)$. By division algorithm, we have,

$p(x)=q(x)d(x)+r(x)\implies p(x)=\left(q(x)\cdot\large\prod_{k=1}^{q_d}(x-r_k)\right)+r(x)$

Obviously, $r(x)$ is a polynomial of degree $q_d-1$ and hence is of the form $\displaystyle\sum_{k=0}^{q_d-1}a_kx^k$ where the sequence $\{a_k\}_{k=0}^{k=q_d-1}$ is a sequence of constants depending upon $d(x)$. We then obtain,

$\large p(r_k)=r(r_k)=\sum_{j=0}^{q_d-1}a_j(r_k)^j~\forall~k\in\Bbb{Z^+_{\leq q_d}}$

So, from the above result, you get $q_d$ equations for a system with $q_d$ unknowns. Hence, you can bash it out using methods of solving large system of equations, mostly methods from Linear Algebra (matrices, etc).

There's yet another way to finish it off elegantly if the roots of $d(x)$ form an arithmetic progression. The above result also can be said to give us $q_d$ different polynomial values for $r(x)$ which has degree $q_d-1$. You can use method of differences now to get the final difference column value easily and then use the "reconstruction formula" in method of differences to get the polynomial $r(x)$, which is your answer. And, we are done. $_{\square}$

Note that the form we took for $d(x)$ accounts for all cases since any other kind of structure of $d(x)$ can easily be obtained by multiplying a scalar to $q(x)$ in the division algorithm which doesn't affect $r(x)$.

Since I already wrote so long for this, let's take the time to state an explicit example.

Example: Find the remainder when $p(x)=x^{10}$ is divided by the cubic polynomial $d(x)=(x-1)(x-2)(x-3)$.

Solution: We have, through our procedure as shown above that,

$p(1)=r(1)=1^{10}=1\quad\textrm{and}\quad p(2)=r(2)=2^{10}=1024\quad\textrm{and}\quad p(3)=r(3)=3^{10}=59049$

Now, construct the difference table for $r(x)$ as follows:

$\begin{array}{|c|c|c|c|c|}\hline x&r(x)&D_1(x)&D_2(x)\\ \hline 1&1&1023&57002\\ 2&1024&58025 \\ 3&59049\\ \hline\end{array}$

Now, we do the reconstruction using the following formula:

$r(x)=r(1)+\sum_{i=1}^2\left(\frac{D_i(1)}{i!}\cdot \prod_{j=1}^i(x-j)\right)=28501x^2-84480x+55980$

As you can infer, this becomes more and more tedious as the degrees of $p(x)$ and $d(x)$ increase, even using method of differences or other interpolation methods.

Note: This method works only for those $d(x)$ which has no repeated roots. As of now, this is not much of a generalization since it doesn't account for cases when $d(x)$ has a repeated root.

. Let me try : Let the divisor be $ax^2+bx+c$ whose roots are $u,v$. So its as if , $P(x)$ is divided by $(x-u)(x-v)$.Since the divisor has a degree of $2$ , the remainder must have degree of 1 and thus it has the form : $kx+m$.By division algorithm we can write :

$P(x)=Q(x) . (x-u)(x-v) + kx+m$

When $P(x)$ is divided by $x-u$ , the remainder is $P(u)$ and When $P(x)$ is divided by $x-v$ , the remainder is $P(v)$. So , to find the required remainder we must solve the following system of equations:

$P(u)=ku+m \\ P(v)=kv+m$

Cheers!

@Nihar Mahajan @Manish Dash @Rajdeep Dhingra @Prasun Biswas

How did this question come to your mind! It was my doubt! Thanks for raising this question, mehul!

For an explicit example, see Remainder Factor Theorem - Intermediate.

After you understood what to do, could you add in a paragraph of explanation?