The Polynomial Remainder Theorem states that the remainder of the division of a polynomial $f(x)$ by a linear polynomial $x - r$ is equal to $f(r)$. In particular, $x-r$ divides $f(x) \iff f(r)=0$

But what if the divisor is not linear and of a degree higher than one?

Consider this question:

Let $\mathcal{P}(x)$ be any polynomial. When it is divided by $(x-13)$ and $(x-17)$, then the remainders are $15$ and $35$ respectively. The remainder, when $\mathcal{P}(x)$ is divided by $(x-13)(x-17)$, is

How I approached it:

$$\mathcal{P}(13)=15\tag1$$ $$\mathcal{P}(17)=35\tag2$$

But how do I figure out the remainder if the degree of the divisor is greater than one?


Alternative hint: $$ \mathcal{P}(x)= g(x)(x-13)(x-17) + ax + b $$ for some $\ a\ $ and $\ b\ $. Your recovered values of $\ \mathcal{P}(13)=15\ $ and $\ \mathcal{P}(17)=35\ $ give you two linear equations to solve for $ a\ $ and $\ b\ $.


Hint: Use Lagrange interpolation to find the remainder, which has degree at most 1.


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