Radiation Therapy and Docetaxel in Treating Patients With HPV-Related Oropharyngeal Cancer

Inclusion Criteria: * PRE-REGISTRATION * Provide written informed consent * Submission of research blood draw to be stored until after surgical resection of the primary tumor and confirmation of human papilloma virus (HPV) positivity (Mayo Clinic Rochester patients only) * Patients with oropharynx carcinoma with a smoking history of ˂ 10 pack-year or equivalent 10 year history of tobacco product use and no recent history (within last 5 years) of tobacco use * REGISTRATION * Histological confirmation of HPV+ squamous cell carcinoma of the oropharynx; HPV positivity will be defined as positive staining for p16 on immunohistochemistry (IHC) * Gross total surgical resection with curative intent of the primary tumor and at least unilateral neck dissection within 7 weeks of registration * Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0 or 1 * Smoking history \< 10 pack years or equivalent 10 year history of tobacco product use * Absence of distant metastases on standard diagnostic work-up =\< 10 weeks prior to registration; (chest computed tomography \[CT\], chest x-ray \[CXR\], positron emission tomography \[PET\]/CT, etc.) * Must have one of the following risk factors: * Lymph node \> 3 cm * 2 or more positive lymph nodes * Perineural invasion * Lymphovascular space invasion * T3 or microscopic T4a primary disease * Lymph node extracapsular extension * Absolute neutrophil count (ANC) \>= 1500/mm\^3 * Platelet count \>= 100,000/mm\^3 * Hemoglobin …