Perhaps the most important problem in representation theory in the 1970s and early 1980s was the determination of the multiplicity of com- position factors in a Verma module. This problem was settled by the proof of the Kazhdan-Lusztig Conjecture which states that the multiplicities may be computed via Kazhdan-Lusztig polynomials. In this paper, we introduce signed Kazhdan-Lusztig polynomials, a variation of Kazhdan-Lusztig polyno- mials which encodes signature information in addition to composition factor multiplicities and Jantzen filtration level. Careful consideration of Gabber and Joseph's proof of Kazhdan and Lusztig's recursive formula for computing Kazhdan-Lusztig polynomials and an application of Jantzen's determinant for- mula lead to a recursive formula for the signed Kazhdan-Lusztig polynomials. We use these polynomials to compute the signature of an invariant Hermitian form on an irreducible highest weight module. Such a formula has applications to unitarity testing. 1.1. The Unitary Dual Problem. In the 1930s, I.M. Gelfand introduced a broad programme in abstract harmonic analysis which would permit the transfer of dif- ficult problems in areas as distinct from analysis as topology to more tractable problems in algebra. Fourier analysis is just one incarnation of this programme. An unresolved component in Gelfand's programme is the classification of the irre- ducible unitary representations of a group, known as the unitary dual problem. In the case of a real reductive Lie group, the problem is equivalent to identifying all irreducible Harish-Chandra modules which admit a positive definite invariant Hermitian form. As Harish-Chandra modules may be constructed via an algebraic method introduced by Zuckerman in 1978 known as cohomological induction, it is of interest to study signatures of invariant Hermitian forms on cohomologically induced modules and to understand how positivity can fail. Cohomological induction is a two-step process in which we compose an induction functor with a Zuckerman functor i . The intermediate module in cohomological induction is a generalized Verma module which admits an invariant Hermitian form if the module to which induction was applied admits an invariant Hermitian form. Formulas for the signatures of invariant Hermitian forms on these intermediate modules may be used to compute signatures of forms on corresponding cohomo- logically induced modules (eg. (Wal84)). This motivates the study of invariant Hermitian forms on Verma modules (see (Wal84), (Yee05)) and irreducible highest weight modules.