Which is the order of the roots z=1 and z=-1 in the equation...

Which is the order of the roots $z=1$ and $z=-1$ in the equation $$ z^5-z^4-2 z^3+2 z^2+z-1=0 ? $$ DEFINITION. One calls the derivative of the polynomial $$ P(z)=a_0 z^n+a_1 z^{n-1}+\ldots+a_k z^{n-k}+\ldots+a_{n-1} z+a_n $$ the polynomial $$ P^{\prime}(z)=a_0 n z^{n-1}+a_1(n-1) z^{n-2}+\ldots+a_k(n-k) z^{n-k-1}+\ldots+a_{n-1} $$ The derivative is usually denoted by the symbol ' (prime).

Which is the order of the roots $z=1$ and $z=-1$ in the equation

$$
z^5-z^4-2 z^3+2 z^2+z-1=0 ?
$$

DEFINITION. One calls the derivative of the polynomial

$$
P(z)=a_0 z^n+a_1 z^{n-1}+\ldots+a_k z^{n-k}+\ldots+a_{n-1} z+a_n
$$

the polynomial

$$
P^{\prime}(z)=a_0 n z^{n-1}+a_1(n-1) z^{n-2}+\ldots+a_k(n-k) z^{n-k-1}+\ldots+a_{n-1}
$$

The derivative is usually denoted by the symbol ' (prime).

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Key Concepts

Multiplicity of a Root

The multiplicity of a root (or the order of a root) of a polynomial is the number of times that the root appears as a factor. In other words, if (z - r) appears k times in the factorization of a polynomial, then r is a root of multiplicity k. A simple root is one with multiplicity 1, while repeated roots have multiplicities greater than 1. This concept is essential in understanding the behavior of polynomials near their zeros and in applications such as solving polynomial equations and analyzing graphs.

Polynomial Derivative and its Role in Determining Root Multiplicity

The derivative of a polynomial provides a useful tool for determining the multiplicity of a given root. The basic idea is that if a polynomial f(z) has a root at z = r of multiplicity k (with k > 1), then not only does f(r) = 0 but also the first k ? 1 derivatives evaluated at r will be zero, with the kth derivative being non-zero. This method allows for systematic verification of whether a given solution is a simple root or a repeated root, thus deeply linking the concept of differentiation with factorization properties of polynomials.

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