Answer:
- f(x) > 0 for (-2.203, -1) U (2.203, ∞)
- f(x) < 0 for (-∞, -2.203) U (-1, 2.203)
Explanation:
A graph shows real zeros at ±2.203 and -1. The value of f(x) changes sign at each of these.
The positive leading coefficient indicates the graph trends generally upward, so the function will be positive in intervals with an even number of zeros to the right.
f(x) positive
There are an even number of zeros to the right of x = 2.203 and the interval (-2.203, -1).
f(x) is positive in the intervals (-2.203, -1) or (2.203, ∞).
f(x) negative
The function will be negative where it is not positive (except at the zeros)
f(x) is negative in the intervals (-∞, -2.203) or (-1, 2.203).
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Additional comment
The graph shows the only rational real root of the function is x=-1. Using synthetic division to reduce the polynomial, we find ...
f(x) = (x +1)(x^4 -3x^2 -9)
The remaining factor is a quadratic in x^2, which has roots ...
x = ±√(1.5 ±√11.25)
The real roots are approximately ±2.20320266118. The value 2.203 used above is a stand-in for the exact value √(1.5 +√11.25)).