3 deleted 2 characters in body
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05AB1E, 10 9 bytes

-1 byte thanks to Erik the Outgolfer

sF³gݨm³OsF³gݨm*O

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Takes x as first argument, n as second and p in ascending order as third.

Explanation

sF³gݨm³OsF³gݨm*O
s         # Forces the top two input arguments to get pushed and swaped on the stack
 F        # Do n times...
  ³        # Push the third input (the coefficients)
   g       # Get the length of that array...
    ݨ     # and create the range [0 ... length]
      m    # Take the result of the last iteration to these powers (it's just x for the first iteration)
       ³**   # Multiply the resuling array with the corresponding coefficients
         O  # Sum the contents of the array
           # Implicit print

05AB1E, 10 9 bytes

-1 byte thanks to Erik the Outgolfer

sF³gݨm³O

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Takes x as first argument, n as second and p in ascending order as third.

Explanation

sF³gݨm³O
s         # Forces the top two input arguments to get pushed and swaped on the stack
 F        # Do n times...
  ³        # Push the third input (the coefficients)
   g       # Get the length of that array...
    ݨ     # and create the range [0 ... length]
      m    # Take the result of the last iteration to these powers (it's just x for the first iteration)
       ³*  # Multiply the resuling array with the corresponding coefficients
         O  # Sum the contents of the array
           # Implicit print

05AB1E, 10 9 bytes

-1 byte thanks to Erik the Outgolfer

sF³gݨm*O

Try it online!

Takes x as first argument, n as second and p in ascending order as third.

Explanation

sF³gݨm*O
s         # Forces the top two input arguments to get pushed and swaped on the stack
 F        # Do n times...
  ³        # Push the third input (the coefficients)
   g       # Get the length of that array...
    ݨ     # and create the range [0 ... length]
      m    # Take the result of the last iteration to these powers (it's just x for the first iteration)
       *   # Multiply the resuling array with the corresponding coefficients
         O # Sum the contents of the array
          # Implicit print
2 added 43 characters in body
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05AB1E, 1010 9 bytes

-1 byte thanks to Erik the Outgolfer

sF³gݨm³*OsF³gݨm³O

Try it online!Try it online!

Takes x as first argument, n as second and p in ascending order as third.

Explanation

sF³gݨm³*OsF³gݨm³O
s          # Forces the top two input arguments to get pushed and swaped on the stack
             (This is in case n is 0 so x is on the stack)
 F         # Do n times...
  ³         # Push the third input (the coefficients)
   g        # Get the length of that array...
    ݨ      # and create the range [0 ... length]
      m     # Take the result of the last iteration to these powers (it's just x for the first iteration)
       ³*   # Multiply the resuling array with the corresponding coefficients
         O  # Sum the contents of the array
           # Implicit print

05AB1E, 10 bytes

sF³gݨm³*O

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Takes x as first argument, n as second and p in ascending order as third.

Explanation

sF³gݨm³*O
s          # Forces the top two input arguments to get pushed and swaped on the stack
             (This is in case n is 0 so x is on the stack)
 F         # Do n times...
  ³         # Push the third input (the coefficients)
   g        # Get the length of that array...
    ݨ      # and create the range [0 ... length]
      m     # Take the result of the last iteration to these powers (it's just x for the first iteration)
       ³*   # Multiply the resuling array with the corresponding coefficients
         O  # Sum the contents of the array
           # Implicit print

05AB1E, 10 9 bytes

-1 byte thanks to Erik the Outgolfer

sF³gݨm³O

Try it online!

Takes x as first argument, n as second and p in ascending order as third.

Explanation

sF³gݨm³O
s         # Forces the top two input arguments to get pushed and swaped on the stack
 F        # Do n times...
  ³        # Push the third input (the coefficients)
   g       # Get the length of that array...
    ݨ     # and create the range [0 ... length]
      m    # Take the result of the last iteration to these powers (it's just x for the first iteration)
       ³*  # Multiply the resuling array with the corresponding coefficients
         O  # Sum the contents of the array
           # Implicit print
1
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05AB1E, 10 bytes

sF³gݨm³*O

Try it online!

Takes x as first argument, n as second and p in ascending order as third.

Explanation

sF³gݨm³*O
s          # Forces the top two input arguments to get pushed and swaped on the stack
             (This is in case n is 0 so x is on the stack)
 F         # Do n times...
  ³         # Push the third input (the coefficients)
   g        # Get the length of that array...
    ݨ      # and create the range [0 ... length]
      m     # Take the result of the last iteration to these powers (it's just x for the first iteration)
       ³*   # Multiply the resuling array with the corresponding coefficients
         O  # Sum the contents of the array
           # Implicit print