We construct a numerical scheme for solving a class of fractional optimal control problems by employing Boubaker polynomials. In the proposed scheme, the state and control variables are approximated by practicing Nth-order Boubaker polynomial expansion. With these approximations, the given performance index is transformed to a function of N + 1 unknowns. The objective of the present formulation is to convert a fractional optimal control problem with quadratic performance index into an equivalent quadratic programming problem with linear equality constraints. Thus, the latter problem can be handled efficiently in comparison to the original problem. We solve several examples to exhibit the applicability and working mechanism of the presented numerical scheme. Graphical plots are provided to monitor the nature of the state, control variable and the absolute error function. All the numerical computations and graphical representations have been executed with the help of Mathematica software.