Polynomial and Polynomial Functions classification and solutions, Cubic, Quartic, Quintic, nth-degree polynomial

We are glad to present you the theory we developed and explained in the book titled: classification Of the polynomial with real coefficients - Application to: Quadratic, Cubic, Quartic, Quintic,...

The Polynomial function f(x) with real coefficients

A polynomial function of degree n in the general form

we can write as

Therefore, the expanded form of (2) is

the source function of the polynomial f(x).

Thus, any polynomial f(x) with real coefficients can be expressed in the translatable form of its source function that is, using shown method we can put every polynomial function back to the origin.

The real roots of a polynomial f(x) with real coefficients

Linear function

Quadratic function

Cubic, quartic, quintic,... functions

Before we proceed to analyze the conditions for the existence of the real roots or the zeroes of the higher degree polynomials let us mention that both, the function-theoretic and the formal algebraic approach to the concept of a polynomial, will be equivalent using the shown method.

The number r is a root of the polynomial f(x) if and only if f(r)=0, or if the polynomial f(x) is divisible by (x-r).

Thus, solving polynomial equation f(x)=0 is equivalent to determining the roots of polynomial function or, finding the roots of a polynomial f(x) amounts to finding its linear divisors.

From the expression (6)

that represents the source form of the polynomial function, we derive the basic classification of

Then, using the results and methods of real analysis we get other types or representatives of the given polynomial function.

The described procedure can, on some way, be compared with many of those known methods used to reduce general or standard form of a polynomial.

Let us compare mentioned reduced forms of the cubic, quartic and quintic polynomial with the basic source forms obtained from the source polynomial (6);

Thus, the classification defines; three types of the cubic functions, ten types of quartic functions, and hundred and sixteen types of quintic polynomial functions, that means, defined are the necessary and the sufficient conditions for each type.

Furthermore, defined are the conditions for existence of real roots, stationary points, as extreme; maximums, minimums, and points of inflection.

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The book is accompanied by the separate eight-page summary to enable quick review (to see,

click the Summary button on the top of the page), and with the computer program on CD-ROM

which proves the use of the theory.

149 pages, 25 examples and 86 figures

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