We　present　a　unified　deterministic　approach　for　experimental　design　problems　using　the　method　of　interlacing　polynomials.　Our　framework　recovers　the　best-known　approximation　guarantees　for　the　well-studied　D/A/E-design　problems　with　simple　analysis.　Furthermore,　we　obtain　improved　non-trivial　approximation　guarantee　for　E-design　in　the　challenging　small　budget　regime.　Additionally,　our　approach　provides　an　optimal　approximation　guarantee　for　a　generalized　ratio　objective　that　generalizes　both　D-design　and　A-design.　Copyright　©　2025　by　SIAM.