For two polynomials $f_1=(x-a)(x-b), f_2=(x-a)(x-e)$, if we add them together: $f_3=f_1+f_2$, $f_3$ only has an integer root that is $a$. I observed that it'd possible to make $f_3$ have more than one integer root, for example if I do: $f'_3=f_1+c\cdot f_2$, where $c$ is a constant value. Please note that this is true just for some $c$ but not all constant values. I also observed that if we multiply $ f_1 $ and $ f_2 $ by two different polynomials, $f'_1, f'_2$ of the same degree (e.g. degree 2) as $f_1$, such that $f'_1, f'_2$ have no roots in common, we wont have this problem. Therefore, $f''_3=f'_1\cdot f_1 + c\cdot f'_2 \cdot f_2$ has only one integer root that is $a$.
Hence, we cannot find a constant value $c$ to make $f''_3$ have more than one integer root.
My question: 1- Is the above observation correct. 2- If so, how I can prove its correctness ? 3- Would the above observation be true for higher degree polynomials $f_1$ and $f_2$, where $\deg(f_1),\deg(f_2)> 2$