Only real roots!

A root of the given polynomial is a value of $x$ that makes one of the 999 polynomial factors equal to $0$ . Since we are only taking into account real roots, we will figure out which roots are complex by looking at the discriminants of each polynomial.

$\Delta_k = 47^2 - 4k = 2209 - 4k$

Whenever $\Delta_k$ is negative, the quadratic only has complex roots. $4k > 2209$ for all $k > 552$ , so we can disregard everything beyond $k = 552$ .

We will use the quadratic formula:

If $x^2 - 47x + k = 0$ , then $x = \frac{47 \pm \sqrt{2209 - 4k}}{2}$ . Since we are being asked for the product of all of the real roots, let's first take the product of the two roots of each individual polynomial:

$\frac{47 + \sqrt{2209 - 4k}}{2} \cdot \frac{47 - \sqrt{2209 - 4k}}{2} = \frac{47^2 - (\sqrt{2209 - 4k})^2}{4} = \frac{2209 - 2209 + 4k}{4} = k$

Now with all of this work out of the way, the product of all roots is $\prod_{k=1}^{552}k = 552!$

So if the product of all roots can be expressed as $n!$ , $n = \boxed{552}$ .