Two polynomials having the same roots are identical

Question

The polynomials $f$ and $g$ have same roots(no repeated roots) and $\{x : f(x) = 2015\} = \{x : g(x) = 2015\}$ = S. Here S is non-empty. Show that $f = g$.

Here $x$ is a complex number. A little help would be appreciated.

Answer 1

Let there be $n+1$ roots of each $f$ and $g$ and let these roots be $a_{0},a_{1},...,a_{n}$

Hence,

$f(x)=A[\prod_{r=0}^{n}(x-a_{0})]$

$g(x)=B[\prod_{r=0}^{n}(x-a_{0})]$

where $A$ and $B$ are constants.

Since there exists an $x_{0}$ such that $f(x_{0})=g(x_{0})=2015$,

we deduce that:

$A[\prod_{r=0}^{n}(x_{0}-a_{0})]=2015=B[\prod_{r=0}^{n}(x_{0}-a_{0})]$

it follows that,

$A=B$, thus $f$ and $g$ are identical.