voldemort - Why do Harry and Dumbledore need to get the memory from Slughorn?

In Harry Potter and the Half-Blood Prince, Dumbledore orders Harry to retrieve the memory about the talk with Tom Riddle/Voldemort from Slughorn. The memory is said to be needed because it contains valuable information that is needed to defeat him. The result is that they know that Voldemort uses Horcruxes.

Why do they not know this before? If the knowledge about Horcruxes is present in school books (albeit forbidden section) and teachers, then surely some magical academics will already have guessed that Voldemort would try to use them?

Also, if I recall correctly, there is a hint in the first book, "some thought that he was no longer alive enough to be able to really die" which hints a lot at the split soul as a price you pay for being immortal.

Why does Slughorn's memory seem so crucial to Dumbledore?

Answer

I disagree with your premise. They didn't need the true memory to find out that Riddle had Horcruxes; Dumbledore already had pretty solid reasons to believe that he did, even before they got the true memory.

From HBP chapter 17 ("A Sluggish Memory"), when they first visit the (edited) memory, Riddle said:

"Sir, I wondered what you know about... about Horcruxes?"

So Dumbledore already knew, from the edited memory, that Riddle was interested in Horcruxes.

He also already suspected that Riddle had more than one Horcrux. The diary being as much weapon as safeguard, Voldemort saying he had "gone further than anybody along the path that leads to immortality", the fact that his appearance had grown less human over the years: all these led Dumbledore to strongly suspect that Riddle had multiple Horcruxes (as he explains in chapter 23, "Horcruxes").

But he still needed the true memory. Why? To find out how many Horcruxes Riddle had. That's the piece Dumbledore didn't know (and couldn't even make a good guess at), and the piece without which they couldn't possibly defeat him. The Horcruxes had to be destroyed first, and that couldn't be done if Riddle suspected they were hunting Horcruxes; the good guys had to know how many Horcruxes there were, or the battle was lost before it started. That's why Dumbledore needed the memory: to find out exactly how many Horcruxes Riddle had planned.