Thanks, Eilcon, for your advice. I hadn't thought about the strength of the carpet cleaner, so I'll check out those that you suggested.
As for the rest of you...my cat has only pooped on the carpet twice, ever. This started only two weeks ago. It is not consistent. He usually uses his box, but always in the front room which is on the other side of the house from his box. Yes, he is very healthy and active, and no, I haven't changed anything about his living or box arrangements or lifestyle.
As for my cat becoming a stray...I have had my cat (and no other pets) for four years. I love him, but I will not tolerate an animal that pees or poops in my living space, or scratches on my things. Period. Should my cat's behavior continue, however, he will quickly find himself outside and no longer be allowed in the house. Period. He will then be cared for as an outdoor cat. Period.
I came to this site looking for a solution to the problem, thinking that cat lovers had experience and would know best. My intent is not to be cold hearted or irresponsible, so please respect that I have certain living expectations that includes not allowing my children to live in an unsanitary environment. I have children who play on that same carpet and I'll always choose the health of my children over a pet.