Can a $5$-th class polynomial have only one solution? because that example: $$x^5 - 3x^4 + 17x^3 - 12x^2 - 11x - 5 = 0$$

I mean that it"s not crucial for every saturday grade polynomial to have actually seven solutions. There may be only one or three. The same for a sixth grade polynomial, there might be only two solutions.

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If this is true, then how I deserve to decide if a 5th grade polynomial has actually only one systems or three and not 5 solutions?

A 5th degree polynomial (with actual coefficients) contends least 1 actual root.A polynomial that odd level has at least 1 real root.

The fundamental theorem the algebra says that a polynomial roots equal to that degree. However, they may be complex and they may be root of multiplicity.

$x^2-1$ has 2 genuine roots.$x^2+1$ has actually 2 complex roots.$x^2-2x + 1$ has one root of multiplicity 2.

Does this help?

Here it is a an easy algorithm because that generating a many polynomials with level $5$ and also a solitary real root:

Take a second-degree polynomial $p(x)$Consider the fourth-degree polynomial $q(x)=p(x)^2$Take part $C\in\ptcouncil.netbbR,D\in\ptcouncil.netbbR^+$ and define $Q(x)$ as $C+D\int_0^xq(t)\,dt.$

$Q(x)$ is a fifth-degree polynomial v a single real root, due to the fact that it is a continuous, weakly increasing and also unbounded (in both directions) function over $\ptcouncil.netbbR$. For instance, v the choices $p(x)=x^2+3$, $C=0,D=5$ us get$$Q(x) = x^5+10x^3+45x$$whose just real source lies in ~ $x=0$.

The main algorithm for counting the number of real zeroes the a polynomial is given by Sturm"s theorem, however yet the computation the the discriminant provides you part information.

Fundamental theorem of Algebra

Every non-zero, single-variable, level $n$ polynomial with complex coefficients has, counted with multiplicity, precisely $n$ complicated roots.

Yes. The polynomial

(x - r)^5 has actually one root r with multiplicity 5.

In general one deserve to construct a polynomial ( role ) with wanted properities

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Is there an analytical way to uncover the number of real positive roots that polynomials that the form $ax^m + bx^2 + x -c = 0$?
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