Ugly artefacts when using Pathfinder - Divide

[SdS - StudioDavidSilva.nl]

This morning I have been trying (in vain) to divide three overlapping circles in separate shapes with the help of Pathfinder - divide. But every time I do this, Illustrator shows ugly artefacts everywhere around the edges of the separate shapes. This is clearly visible when you drop the fill color opacity. Does anyone know how to prevent this from happening, or does anyone now how to remove these artefacts?

[Doug A Roberts · Accepted solution]

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[SdS - StudioDavidSilva.nl]

Thank you Doug! You're a life saver: my sanity thanks you!!