Solution:

The coefficient of $x^k$ in $P(x)$ is $24 - |24 - k|$ for $1 \le k \le 47,$ and consecutive coefficients differ by $+1$ up through $x^{24}$ and by $-1$ afterwards. Multiplying by $1 - x$ therefore telescopes: $$(1 - x)P(x) = (x + x^2 + \cdots + x^{24}) - (x^{25} + \cdots + x^{48}) = (x + x^2 + \cdots + x^{24})(1 - x^{24}),$$ so for $x \ne 1,$ $$P(x) = x\left(\frac{x^{24} - 1}{x - 1}\right)^2.$$

The distinct zeros of $P$ are therefore $0$ together with the $24$th roots of unity other than $1:$ $z_k = \cos 15k^\circ + i \sin 15k^\circ$ for $k = 1, \ldots, 23.$ The zero $0$ contributes nothing, and $z_k^2 = \cos 30k^\circ + i \sin 30k^\circ,$ so $|b_k| = |\sin 30k^\circ|.$

As $k$ runs from $1$ to $12,$ the values $|\sin 30k^\circ|$ are $\frac{1}{2}, \frac{\sqrt{3}}{2}, 1, \frac{\sqrt{3}}{2}, \frac{1}{2}, 0$ repeated twice, summing to $4 + 2\sqrt{3};$ the terms for $k = 13, \ldots, 23$ repeat those for $k = 1, \ldots, 11$ and add another $4 + 2\sqrt{3}.$ The total is $8 + 4\sqrt{3},$ so $m + n + p = 8 + 4 + 3 = 15.$