Classical and New Inequalities in Analysis 3.

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## Polynomials and Polynomial Inequalities, Springer-Verlag

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Like any other function , a polynomial may be written as an inequality, giving a large range of solutions. The best way to solve a polynomial inequality is to find its zeros. The easiest way to find the zeros of a polynomial is to express it in factored form. At these points , the polynomial's value goes from negative to positive or positive to negative. This knowledge can then be used to determine the solutions of the inequality.

## 11.09. Polynomial Inequalities

Much of the work involved with solving inequalities is based in observation and judgement of a particular mathematical situation, and is therefore best demonstrated with an example. This can be expressed as the product of three terms :. From the explanation above, we know that the polynomial is positive as it approaches its next zero, but we can use the same reasoning for proof.

Recalling the initial inequality, we can now determine the solution of exactly where the polynomial is greater than zero. This polynomial has three roots. It is positive in two segments and also negative in two. If it were a polynomial inequality with the condition that all values are greater than zero, the two negative segments would be removed.

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