I recently notice google is indexing my site one with /home and one without. clearly this www.test.come/home URL exist but i do not see this url in my URL rewrite in the back-end. what is the best solution to remove this /home so google is not indexing the home pages
Yes. There will be some variants available for your home page of store. In case you have SSL there will be more 2 variants.
You can add this code in header to tell Google to use it as canonical url:
<link rel="canonical" href="http://yourdomain.com">