This study shows that a controllable systemẋ = Ax + Bu, wherexis ann-vector, can be driven from an arbitrary initial conditionx(0) = x0to an arbitrary final conditionx(tf= xfby a polynomial control function of degreeM= 2r+ 1, wherer=n− rank (B), through a polynomial trajectory of degreeM. A simple algorithm for findinguby solving a set of linear equations, the solution of which yields the polynomial coefficients, is given.