EXAMPLE 1. Determine the number and situation of the real roots of
the equation x^5 - 3 x - 1 = 0.
The Sturm chain is
y = x^5 - 3 x - 1,
y' = 5 x^4 - 3,
y'' = 12 x + 5,
y''' = 1.
where y'= dy/dx
The signs of f for x = -2, -1, 0, +1, +2 are
-----------------------
x | f0 | f1 | f2 | f3
-----------------------
-2 | - | + | - | +
-1 | + | + | - | +
0 | - | - | + | +
+1 | - | + | + | +
+2 | + | + | + | +
-----------------------
The equation thus has three real roots: one between -2 and -1, one
between -1 and 0, one between +1 and +2. The other two roots are complex.
Question:
How do you know that the equation has three real roots between -2 and -1, one
between -1 and 0, one between +1 and +2???
Please explain!!
2007-01-11
00:43:11
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1 answers
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asked by
Anonymous
in
Mathematics