If I edit an MP3 in GoldWave, or chop off bits and pieces, will it reduce the quality of the MP3 further, or will it remain the same?
When you load an MP3 into Goldwave, it appears as the waveform you see. Editting the waveform or making any changes will NOT directly affect the quality...
BUT resaving the file as an mp3 will result in a loss of quality compared to the first mp3.
Why? Think of it like a photocopier. Think of "Saving to an mp3" as "making a copy" in a photocopier. A copy is never quite as good as the source data being copied. The more times you "copy a copy in a photocopier" (the more times you save an mp3 to another mp3), the more quality you lose.
The CD gets "Copied" to the first .mp3, which you load into Goldwave. It doesn't matter what you do with this file in Goldwave. What makes all the difference is how you SAVE the new file. If you save as an mp3 again, it's the equivalent of making a copy of a copy.
The question is, how negligible is this loss in quality? If you're saving from a 320kbps mp3 source tile to another 320 kbps mp3, you will lost *almost* nothing. The result will probably sound like a 310 kbps or similar....though the new file will still have the equal Quantity of data as the original, meaning you are wasting space by doing this.
In a perfect world in order to preserve every bit of quality, you need to do this:
Rip the CD to a .wav file.
Load the .wav into Goldwave
Make your changes
Save to mp3 (bitrate of your choice).
This way you're only using the "photocopier" once and not making copies of copies. B) That being said though, that's theoretical. If you're using high bitrates (192kbps or more), chances are you won't hear a drop in quality unless you're a supreme audiophile.

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So the short answer to your question:
will it reduce the quality of the MP3 further, or will it remain the same?
is "It will reduce the quality of the mp3 further, yes."