The author defines the Jacobsthal polynomials \(\{J_n(x)\}\) and the Jacobsthal-Lucas polynomials \(\{j_n(x)\}\) recursively by \[ J_{n+2} (x)=J_{n+1} (x)+2xJ_n (x),\;J_0(x)=0,\;J_1(x)=1, \]
\[ j_{n+2}(x)= j_{n+1}(x)+ 2xj_n(x),\;j_0(x)=2,\;j_1(x)=1. \] Obviously, \(J_n(1/2)=F_n\) and \(j_n(1/2)=L_n\). For \(x=1\), the polynomials reduce to the Jacobsthal and Jacobsthal-Lucas numbers. The author obtains summation formulas, explicit closed forms and several identities for these polynomials. He also studies the diagonal functions and the “augmented” polynomials generated by them.