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#Java 8, 202 176 174 bytes

a->{String r="";int j=a.length;for(int i:a)r+=i==0*j--?"":"+"+i+(j<1?"":j<2?"x":"x^"+j);return r.isEmpty()?"0":r.substring(1).replace("+-","-").replaceAll("([+-])1x","$1x");}

• 26 bytes thanks to @Nevay.

Explanation:

Try it online.

a->{                     // Method with String-array parameter and String return-type
  String r="";           //  Result-String, starting empty
  int j=a.length;        //  Power-integer, starting at the size of the input-array
  for(int i:a)           //  Loop over the array
    r+=i==0              //   If the current item is 0
           *j--?         //   (And decrease `j` by 1 at the same time)
        ""               //    Append the result with nothing
       :                 //   Else:
        "+"              //    Append the result with a "+",
        +i               //    and the current item,
        +(j<1?           //    +If `j` is 0:
           ""            //      Append nothing more
          :j<2?          //     Else-if `j` is 1:
           "x"           //      Append "x"
          :              //     Else:
           "x^"+j);      //      Append "x^" and `j`
  return r.isEmpty()?    //  If `r` is still empty
    "0"                  //   Return "0"
   :                     //  Else:
    r.substring(1)       //   Return the result minus the leading "+",
     .replace("+-","-")  //   and change all occurrences of "+-" to "-",
     .replaceAll("([+-])1x","$1x");}
                         //   and all occurrences of "1x" to "x"