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 \>= 9.0 g/dL * Direct bilirubin within upper limit of normal (ULN) * Creatinine =\< ULN x 1.5 * Negative pregnancy test done =\< 7 days prior to registration, for women of childbearing potential only * Ability to complete questionnaire(s) by themselves or with assistance * Provide informed written consent * Willingness to return to enrolling institution for follow-up (during the active monitoring phase of the study) Exclusion Criteria: * Any significant tobacco history within the past five years * Any of the following: * Pregnant women * Nursing women * Men or women of childbearing potential who are unwilling to employ adequate contraception * Co-morbid systemic illnesses or other severe concurrent disease which, in the judgment of the investigator, would make the patient inappropriate for entry into this study or interfere significantly with the proper assessment of safety and toxicity of the prescribed regimens * Immunocompromised patients and patients known to be human immunodeficiency virus (HIV) positive * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements * Receiving any other investigational agent which would be considered as a treatment for the primary neoplasm * Other active malignancy =\< 5 years prior to registration; EXCEPTIONS: non-melanotic skin cancer or carcinoma-in-situ of the cervix; NOTE: if there is a history or prior malignancy, they must not be receiving other specific treatment for their cancer * History of myocardial infarction =\< 180 days prior to registration, or congestive heart failure requiring use of ongoing maintenance therapy for life-threatening ventricular arrhythmias * Prior history of radiation therapy to the affected site * History of connective tissue disorders such as rheumatoid arthritis, lupus, or Sjogren's disease * Presence of any of the following risk factors after surgery: * Any positive surgical margin * Adenopathy below the clavicles * Prior systemic chemotherapy for the study cancer; NOTE: prior chemotherapy for a different cancer is allowable * History of allergic reaction to docetaxel * Receiving any medications or substances that are strong or moderate inhibitors of cytochrome P450, family 3, subfamily A, polypeptide 4 (CYP3A4) * Use of strong or moderate inhibitors is prohibited =\< 7 days prior to registration * Receiving any medications or substances that are inducers of CYP3A4 * Use of inducers is prohibited =\< 12 days prior to registration