Positivity Conditions for Cubic and Quintic Polynomials

We present a necessary and sufficient condition for a cubic polynomial to be positive for all positive reals. We identify the set where the cubic polynomial is nonnegative but not all positive for all positive reals, and explicitly give the point where the cubic polynomial attains zero. We then reformulate a necessary and sufficient condition for a quartic polynomial to be nonnegative for all positive reals. From this, we derive a necessary and sufficient condition for a quartic polynomial to be nonnegative and positive for all reals. Our condition explicitly exhibits the scope and role of some coefficients, and has strong geometrical meaning. We also state the geometric features of the nonnegativity region and the positivity region. Finally, by using the Sturm sequences, we present a necessary and sufficient condition for a quintic polynomial to be positive and nonnegative for all positive reals.