Algebra Practice #2

$\frac{x^{24}}{24}+8 \cdot 3^{11}$ can be written as $k \cdot A(x) \cdot B(x) \cdot C(x) \cdot D(x)$, where $k$ is a rational number and $A(x), B(x), C(x)$ and $D(x)$ are non-constant polynomials with rational coefficients. Find the number of terms (including $k$) in the new form.

For example, the polynomial $\frac{4}{63} \cdot (x^2+4x+3) \cdot (x+2) \cdot (x^6+4x^5) \cdot (9x^3+5)$ would have $10$ terms.