I think all the existing answers missed the most important point. We know that Dumbledore already knew about the Horcruxes because Dumbledore had already tracked down and destroyed one of them!
Before the school year had even started Dumbledore had already researched enough into Voldemort's past to know that he had created Horcruxes and that one of them (the ring) was hidden in the ruins of the house of Voldemort's relatives.
In fact, Dumbledore tells Harry that he knew of the Horcruxes well before this. In one of their sessions in Half-Blood Prince he says:
"Four years ago, I received what I considered certain proof that Voldemort had split his soul."
He then goes on to explain how the diary that Harry had found in Chamber of Secrets was clearly a Horcrux.
Later in that same conversation Dumbledore even says that he knew that Voldemort's goal had been to make multiple Horcruxes:
"The careless way in which Voldemort regarded this Horcrux seemed most ominous to me. It suggested that he must have made — or been planning to make — more Horcruxes, so that the loss of his first would not be so detrimental.I did not wish to believe it, but nothing else seemed to make sense.
Then Dumbledore tells Harry of further evidence that Voldemort had made multiple Horcruxes:
"Then you told me, two years later, that on the night that Voldemort returned to his body, he made a most illuminating and alarming statement to his Death Eaters. 'I, who have gone further than anybody along the path that leads to immortality.' That was what you told me he said. 'Further than anybody,' And I thought I knew what that meant, though the Death Eaters did not. He was referring to his Horcruxes, Horcruxes in the plural, Harry, which I do not believe any other wizard has ever had.
So, what we see from this is that Slughorn's memory was decidedly not needed to clue in Dumbledore to the multiple Horcruxes. Had Dumbledore never seen the true memory he would still have hunted Horcruxes, as in fact he had been doing before he saw the real memory.
However, there were still some unknown factors. Knowing that Slughorn had a tampered memory involving a discussion with Voldemort about Horcruxes is surely enough of a reason to want to see it, especially if you already know that Voldemort had created multiple Horcruxes. The memory might contain any number of useful bits of information that might help in the fight against Voldemort. Indeed, it did contain some useful information, namely, that Voldemort had the idea that seven Horcruxes would be especially powerful.
Dumbledore may or may not have suspected that the true memory contained information specifically pertaining to the number of Horcruxes, but he certainly suspected that it would have some useful information. In fact, after they view the tampered memory, Dumbledore explicitly states that he doesn't know how important the true memory will be:
It is most important that we secure the true memory, Harry.... How important, we will only know once we have seen the real thing.