How did you create the pentagon?
Could you please post a screenshot taken at View > 100% and with the pertinent Panels (Paths, Channels, Options Bar, …) visible?
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Refine edge is for modifying the edges of selections. It is particularly useful where those edges contain fur of fine hair. It is not for smoothing our vectors.
If you use the shape tool and stroke the edge (as shown above by c.pfaffenbichler ) you will get the smoothest stroke which will display at your chosen pixel resolution. Because a shape is a vector, scaling will not reduce the resolution.
However, as with any image being displayed, if you zoom in greater than 100% you will start to see the pixels. This is unavoidable as the shape cannot be displayed at a higher resolution than your canvas.
Zoom in :
However - because this is a vector based shape - if I increase the Image size by 400% and then zoom out again to display it at 100% - I get smooth edges again