We get this question a lot from our clients and although this was a good strategy 5 year ago, Google is now penalizing websites for doing this. :(
Rankings in search engines used to be heavily influenced by keywords inside the domain name, and this created a frantic mindset to buy domains with keywords in them. While it is still a minor influence, it is nothing like it used to be.
On the other hand if the domains are brand new and they've never resolved to a site before, then there won't be any kind of impact for SEO.
The main problem is that Google hates duplicate content, so if you have several addresses that point to the same content, Google will see that more than one website has the same content and they will devalue the ranking on all the sites that copied it. The only way to make those other domain names come up on the first page of Google would be to promote each one of them like you do with your website.
The only time I would recommend using multiple domains is if you have a lot of resources (time and money) to create LANDING PAGES outside of your website to drive traffic to your site. These landing pages would have to be separate sites with ONE function, to generate leads about a specific service and they have to be VERY specific about it. The only issue with this is that it is very time consuming: it takes too much time to create, update and promote these pages. This is a strategy that larger businesses use because they have the resources.
Some people will tell you this works, but that does not mean it will work for you and it definitely does not mean it is the right thing to do in Google's eyes, and that is what’s most important. We cannot trick Google, there is a limit and this strategy is on the other side of that limit.
In the SEO world this is called BLACK HAT SEO. If you do Black Hat you can get FAST RESULTS but you have a higher probability to loose them all very fast too. This is what happened with the past 2 updates Google did to their algorithm (Panda and Penguin). A lot of websites lost their entire ranking because they were trying to trick Google with repeated content, multiple domains and inlinks from weird websites.