If I say 'Please tell me if you *would* know this', does it sound more polite than without "would"?

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Perhaps it’s better if you switched the sentence up a bit. You could try: Would you please tell me if you know this? But yes it does sound polite!

No. It changes the meaning of the sentence.

it would sound better without the word "would"

@polychromatic Using "would" like that sets up a conditional clause (an if/then clause).
"Please tell me if you would know this IF..."
"Please tell me if you would know this if the question came up on the exam?"
"Please tell me if you would know this song if it came on the radio?"

You're not asking "Do you know this?" You're asking "Under a specific set of circumstances, would you know this?"

That said, you can ask simply "Would you know this?"
"Would you know the answer to number 3?"
"Would you know the name of that Adele song?"
Technically, this is also sets up for a conditional clause, but in this context, it's like asking "Would you happen to know this?"
It's a way of beating around the bush to make a question sound less forward.
But it doesn't work when proceeded by "Please tell me..."