# Does Polynomial Remainder Theorem work with divisors that are quadratic?

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\$$.