We investigate the $J_1$-$J_2$ spin chain consisting of spins with magnitude $\frac12$. The nearest-neighbor and the next-nearest-neighbor exchange interactions are ferromagnetic and antiferromagnetic, respectively, and induce strong frustration. Both these interactions involve the bond alternation. We find exact solutions for all the degenerate ground states on the phase boundary of the ferromagnetic phase. The degeneracy remains irrespective of two parameters representing the bond alternation. The exact solutions are of closed forms for no bond alternation and of recursion formulae in general. The exact solutions are applicable to the $\Delta$ chain as a special case.