[math-fun] Polynomials with real roots

Stupid matrix question: I know that the characteristic polynomial of a Hermitian matrix must have all real roots (it is diagonalizable into a real diagonal matrix). But is every polynomial with only real roots the characteristic polynomial of a Hermitian matrix? If so, is there a construction that takes one from the polynomial (in the form of a vector of coefficients) to the Hermitian matrix? (The construction from the roots themselves is trivial: they form a diagonal matrix that can be rotated by any unitary matrix.) Presumably, such a construction would fail if/when not all of the roots are real.