<s>[INST] Polynomials division Let $p$ be a real polynomial ($p\in\mathbb{R}[x])$, if 


*

*5 is the remainder of the division $\frac{p(x)}{x-2}$, and

*2 is the remainder of the division $\frac{p(x)}{x-5}$


What is the reaminder of the division $\frac{p(x)}{(x-2)(x-5)}$?
I have tried it, but I have no idea... 
Thank you
 [/INST] $p(x)=(x−2)(x−5)q(x)+ax+b$, for some $a,b \in \mathbb{R}$
Since we know the remainder of $p(x)$ when divided by $x-2$ and $x-5$
we obtain the following
\begin{align}
p(2) &= (2-2)(2-5)q(2) + a(2)+b\\ &=  2a+b = 5  \label{a}
\end{align}
\begin{align}
p(5) &= (5-2)(5-5)q(5) + a(5)+b\\ &=  5a+b = 2 \label{b}
\end{align}
Solving the simultaneous equations \ref{a} and \ref{b}, we obtain $a = -1 \text{ and } b = 7$.
Hence, the remainder of the division $\frac{p(x)}{(x−2)(x−5)}$ is $-x + 7$.
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