[Nancyx]

Hi.

I can’t figure out the systematic name of this polymer.

Thank you.

[MNIO]

naming polymers is not for the faint of heart 

notice your repeating unit
             CH3
             |
  - [-CH-CH-]-
             |
             CH
             ||
             CH2

we polymerize across things like double bonds so we would expect the monomer to be something like
         CH3
         |
 CH2=C
         |
         CH
         ||
         CH2

this is 2-methyl-1,3-butadiene.  aka "isoprene"

if you have trouble finishing naming this critter.. try the wikipedia link
https://en.wikipedia.org/wiki/Polyisoprene

[AWK]

Rather, as a polymer it is a derivative (constitutional unit CU) of 1-ethenyl-1-methylethylene or 1-methyl-1-vinylethylene.
https://iupac.org/wp-content/uploads/2016/06/preferred-names-of-constitutional-units.pdf

[MNIO]

AWK.. Thanks for the IUPAC table.  I've saved it in my reference material. 

I'm not sure I'd write "rather as a polymer it is a derivative (constitutional unit CU) of 1-ethenyl-1-methylethylene or 1-methyl-1-vinylethylene" instead of "polyisoprene 1,2-addition" but to each his own.  Point was to given Nancyx some direction.  I think we've done that.

cheers

[Nancyx]

Thank you for your answers MNIO and AWK .
Would 1,2-polyisoprene be a correct systematic name?

[AWK]

The nomenclature is slowly adjusting to what chemists imagine and then get in the laboratory. Names like 1,2-polyisoprene (1,2-PIP) or 3,4-polyisoprene (3,4-PIP) have appeared in the literature but are not systematic names yet.

[Nancyx]

OK, then what about 1,2-poly-2-methyl-1,3-butadiene?

[MNIO]

notice that on page 31 of that IUPAC link AWK gave us, there are multiple columns for naming polymers under IUPAC.
  column 2 = "structural based polymer name"
  column 3 = "source based polymer name". 
  column 4 = "retained original"
  column 5 = "no longer acceptable"
notice the additional info.
  PIN = Preferred IUPAC Name
  bold face = Preferred among source based polymer names

on page 33 column 2 (structural based) you see
  poly(1-ethenyl-1-methylethylene)
  poly(1-methyl-1-vinylethylene) 1,2-polymerization

on page 33 column 3 (source based) you see
  polyisoprene⁵
  poly(2-methylbuta-1,3-diene)

you can see that both those source based names (without other info) apply to all the different variations of the polymer (1,4 & 1,2 & 3,4)

notice the ⁵ in polyisoprene⁵.  At the end of the table, you'll see this footnote
⁵ Polybutadiene (polyisoprene) can be used as name for a polymer made from butadiene (isoprene), if the structure is not known.

**********
none of the possible names for your compound have PIN denotation so you can use either
  poly(1-ethenyl-1-methylethylene)
  poly(1-methyl-1-vinylethylene) 1,2-polymerization

and I would think
  polyisoprene 1,2-polymerization
would work just as well

*********
As to which you'll find in industry.. "isoprene" has been around for a couple of hundred years now.  It's a well known name so much so that if it weren't in the "source based name" (column 3), it would be in the "Retained Original" name, column 4. 

and furthermore, if you try to buy (or sell) this chemical, I think you'd have a hard time with poly(1-ethenyl-1-methylethylene). 

************
you choose.  all are defendable under IUPAC rules.