A question about zeros of polynomials

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In summary, the zeros of a polynomial are the values of the variable that make the polynomial equal to zero. To find these zeros, you can use methods such as factoring, the quadratic formula, or using a graphing calculator. These zeros are important because they help us understand the behavior and properties of the polynomial function. They are also related to the factors of the polynomial, as they are the solutions to the equation formed by setting the polynomial equal to zero. A polynomial can have multiple zeros, with the number of zeros being equal to its degree.
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zetafunction
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POlynomials (or Taylor series ) of the form

[tex] P(x)= \sum_{n}a_{2n}X^{2n} [/tex] with [tex] a_{2n}\ge 0 [/tex] strictly

have ALWAYS pure imaginary roots ??

it happens with [tex] sinh(x)/x [/tex] [tex] cos(x) [/tex] could someone provide a counterexample ? is there an hypothesis with this name ??
 
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Hi zetafunction! :smile:

Hint: start by treating it as a polynomial in x2, and factor it as (x2 + z1)(x2 + z2)…(x2 + zn).

What happens if any of the zs are imaginary? :wink:​
 

Related to A question about zeros of polynomials

What are the zeros of a polynomial?

The zeros of a polynomial are the values of the variable that make the polynomial equal to zero. In other words, they are the solutions or roots of the polynomial equation.

How do I find the zeros of a polynomial?

To find the zeros of a polynomial, you can use various methods such as factoring, the quadratic formula, or using a graphing calculator. These methods involve manipulating the polynomial equation to solve for the variable.

Why are the zeros of a polynomial important?

The zeros of a polynomial help us understand the behavior and properties of the polynomial function. They can tell us about the x-intercepts of the graph, the number of solutions to the equation, and the degree of the polynomial.

What is the relationship between the zeros of a polynomial and its factors?

The zeros of a polynomial are also the x-intercepts of its graph, which means they are the points where the polynomial crosses the x-axis. These points are also the roots of the polynomial equation, which are the values that make the polynomial equal to zero. The factors of a polynomial are the expressions that when multiplied together, result in the polynomial. Thus, the zeros of a polynomial are the solutions to the equation formed by setting the polynomial equal to zero, and the factors are the expressions that make up the polynomial.

Can a polynomial have more than one zero?

Yes, a polynomial can have multiple zeros. In fact, the number of zeros of a polynomial is equal to its degree. This means that a polynomial of degree n can have up to n distinct zeros.

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