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1 answer:
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as you already know, Descartes rule of signs check the times the sign changes or f(x) and for f(-x)
by the fundamental theorem of algebra, the polynomial has a degree of 5, so it has to have at most 5 zeros/solutions/roots.
f(x) has 3 sign changes, notice, that means, it has either 3, or (3-2), 1 positive zeros.
for f(-x), recall that x³, will be (-x)³ = (-x)(-x)(-x) = -x³, so in short, if the exponent is ODD, the sign changes for that term, if it's EVEN, it doesn't change.
so for f(-x), we have 2 sign changes, meaning, it has either 2 or (2-2), 0 negative roots.
the slack is picked up by the complex roots.
so
3 positive, 2 negative, 0 complex
or
1 positive, 2 negative, 2 complex *recall complex always come in pairs*
or
3 positive, 0 negative, 2 complex
or
1 positive, 0 negative, 4 complex.