A Pair of Basic Inequalities for Polynomials with Real Coefficients

Hence, the process of solving polynomial inequalities involves identifying the function’s zeros and ascertaining the sign of each interval. In order to solve the inequality, it suffices to select a test-point for each interval to determine the sign of the entire interval.


Solution:

By performing some algebraic manipulations, it can be determined that the inequality to be proven is $-6asigma_1le50+15a^2$. Thus, it is necessary for $sigma_1$ to be less than or equal to $-frac {50+15a^2}{6a}$.

However, there is no maximum limit for $sigma _1$, resulting in the general inconsistency of the inequality.

When $a$ is equal to -1 and $sigma_1$ is equal to 20, the inequality does not hold.

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