Proving Three Zeroes for x3 - 15x + 1 in Closed Interval [-4,4]

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Homework Statement


Show that x3 - 15x + 1 has three zeroes in the closed interval [-4,4].


Homework Equations


I think it's just subbing.


The Attempt at a Solution



Well, with that range, I think I can just list them as so:

-4
-3
-2
-1
0
1
2
3
4

From here, I figure out an answer by subbing each into each. And through the intermediate value theorem, if 2 numbers have a 0 between them, that should prove one zero is present. I continue this until I discover 3. So first, I when subbing in -4, I get:

-64 + 50 + 1 = -3

Subbing in -3 for x, we get:

-27 + 45 + 1 = 19

So there must be a 0 between -3 and 19. Thus, I have found one, I continue until I find 2 more?

Am I doing this question right? Thanks for the help!
 
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Discussion

Sure, if you can find three sign changes, then there are at least three roots.
 
Thanks! I guess sign changes is a better way of looking at it. Appreciate the help!
 
Though the sign changes do not necessarily show up at integer values of x.

For example, if the function is 0 at x= 1/2 and at x= 3/4, looking at x= 0 and 1 would miss that.