Full text search through many tables in MySQL
Sure, it's fine to copy the searchable content to your full_text_search table.  

MySQL supports FULLTEXT indexes only in the MyISAM storage engine (until MySQL 5.6, but fulltext in InnoDB in MySQL 5.6 still seems a little unstable).  So you can store your canonical data in InnoDB for safety, and a copy in MyISAM for indexing.  MyISAM is susceptible to data corruption, but if it's only a copy then you just need to repopulate the MyISAM table if it ever gets corrupted.

Your use of master_id as distinct from the primary key of each table is a little strange.  Why not use the primary key, and add another column to your full_text_search table for the type of content?

CREATE TABLE full_text_search
(
    id INT NOT NULL,
    content_type ENUM('news','polls','comments','galleries','articles') NOT NULL,
    content_text TEXT NOT NULL,
    PRIMARY KEY ( id, content_type )
);

full_text_search table:
+----+--------------+---------------+
| id | content_type | content_text  |
+----+--------------+---------------+
|  1 |         news |     ...       |
|  1 |     articles |     ...       |
|  2 |     articles |     ...       |
+----+--------------+---------------+


Another option is to create a fulltext search index in another specialized technology such as Sphinx Search or Apache Solr.  But the same pattern would be useful -- store the primary key field and a field for the type of content.