If a polynomial function f(x) satisfies f(f(f(x)) = 8x+21, where p and q are real numbers, then p+q is equal  to _______

$\begin{array}{l}\mathrm{f}\left(\mathrm{x}\right) \mathrm{must} \mathrm{be} \mathrm{linear} \mathrm{function}. \mathrm{Therefore},\\ \mathrm{f}\left(\mathrm{x}\right)=\mathrm{px}+\mathrm{q}\\ \therefore \mathrm{f}\left(\mathrm{f}\left(\mathrm{x}\right)\right)=\mathrm{pf}\left(\mathrm{x}\right)+\mathrm{q}=\mathrm{p}\left(\mathrm{px}+\mathrm{q}\right)+\mathrm{q}={\mathrm{p}}^2\mathrm{x}+\mathrm{pq}+\mathrm{q}\\ \therefore \mathrm{f}\left(\mathrm{f}\left(\mathrm{f}\left(\mathrm{x}\right)\right)\right)=\mathrm{p}\left({\mathrm{p}}^2\mathrm{x}+\mathrm{pq}+\mathrm{q}\right)+\mathrm{q}\\ ={\mathrm{p}}^3\mathrm{x}+{\mathrm{p}}^2\mathrm{q}+\mathrm{pq}+\mathrm{q}\\ =8\mathrm{x}+21 \left(\mathrm{Given}\right)\\ \therefore {\mathrm{p}}^3=8 \mathrm{or} \mathrm{p}=2\\ \mathrm{and} {\mathrm{p}}^2\mathrm{q}+\mathrm{pq}+\mathrm{q}=2 \mathrm{or} \mathrm{q}=3\\ \therefore \mathrm{p}+\mathrm{q}=5\end{array}$