Significance of a polynomial

From remainder theorem, we get that,

$f(x)$ has a factor of $(x-k)$ for every numbers from $k = 1$ to $4536$ .

$f(x)$ has a factor of $(x-1)(x-2)(x-3)...(x-4536)$ .

But $(x-1)(x-2)(x-3)...(x-4536)$ has degree of 4536, so there must be another polynomial $P(x)$ such that $f(x) = P(x)(x-1)(x-2)...(x-4536)$ and $deg(P(x)) = 1$ .

Substituting $x = 4537$ and $x = 0$ we get

$f(4537) = P(4537)\times 4536!$

$f(0) = P(0)\times 4536!$ .

Subtract each other we get

$f(4537) - f(0) = 4536!(P(4537) - P(0))$

Suppose $P(x) = x - c$ for constant $c$ . ( $f(x)$ is monic, or leading coefficient = 1, so we have coefficient of $x$ is 1)

$P(4537) - P(0) = (4537 - c) - (0 - c) = 4537$ .

Therefore, $f(4537) - f(0) = 4537!$ . ~~~ Ans: $\boxed{4537}$ .

By observation lets say the polynomial is (x-1)(x-2)(x-3).........(x-4536)(x-a)

where a is the 4537th root of the polynomial . Now simply put 4537 and 0 in the polynomial you can see the term containing a gets cancelled up!

THIS IS NOT A SOLUTION

@Aniket Sanghi -Where did you get this question from? Can you tell me the resource?