Find the equation of a polynomal with the following information

Given that $f(5)=5$ and it crosses $y=0$ at the following points:

$x=200, x=100,x=0,x=-100,x=-200$

Since the crossing points of this function is rational, I deduct that the function is factorizable.

Perform reverse-factorization:

The function need to be in this form:

$g(x)=(x+200)(x+100)(x+0)(x-100)(x-200)$

For it to have roots at $x=200, x=100, x=0, x=-100,x=-200$

There are a total of five zero-points at:

$1.x=-200 , y=0$

$2.x=-100 , y=0$

$3.x=0 , y=0\space(Origin)$

$4.x=100 , y=0$

$5.x=200, y=0$

There are a total of four critical points at:

Local Maxima:

$1.x=\sqrt{15000-1000\sqrt{145}} , y=200000000\left(\sqrt{5}+5\sqrt{29}\right)\sqrt{30-2\sqrt{145}}$

$2.x=-\sqrt{15000+1000\sqrt{145}} , y=-200000000\left(\sqrt{5}-5\sqrt{29}\right)\sqrt{30+2\sqrt{145}}$

Local Minima:

$3.x=\sqrt{15000+1000\sqrt{145}} , y=\left(\sqrt{5}-5\sqrt{29}\right)\sqrt{30+2\sqrt{145}}$

$4.x=-\sqrt{15000-1000\sqrt{145}} , y=-200000000\left(\sqrt{5}+5\sqrt{29}\right)\sqrt{30-2\sqrt{145}}$

Since $f(5)=5$, Just divide every y-value of maximas and minimas above by a factor of $g(5)/5=398750625$

$f(x)=\frac{1}{398750625}\left(x+200\right)\left(x+100\right)x\left(x-100\right)\left(x-200\right)$

Q.E.D.

(I bet you just randomly typed the numbers in, didn't you? (Because the $x$ and $y$ values are pretty ugly to be honest))

(After downloading printscreen and photo-editing plugins later...)
(Wait I think I messed the pictures up...)