Fundamental Theorem of Algebra

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What is the Fundamental Theorem of Algebra?

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The Fundamental Theorem of Algebra states that every non-constant polynomial equation of degree n has exactly n roots in the complex number system, counting multiplicities.

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What does the degree of a polynomial indicate?

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The degree of a polynomial indicates the highest power of the variable in the polynomial and determines the maximum number of roots (real and complex) the polynomial can have.

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How many roots does a polynomial of degree 6 have?

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A polynomial of degree 6 has exactly 6 roots in the complex number system, counting multiplicities.

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What is the maximum number of negative roots a polynomial can have?

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The maximum number of negative roots a polynomial can have is determined by Descartes' Rule of Signs, which states that the number of negative roots is equal to the number of sign changes in the polynomial's coefficients.

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If a polynomial has a degree of 5, what is the total number of roots?

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A polynomial of degree 5 has exactly 5 roots in the complex number system, counting multiplicities.

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What is Descartes' Rule of Signs?

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Descartes' Rule of Signs is a method for determining the number of positive and negative real roots of a polynomial based on the number of sign changes in the polynomial's coefficients.

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How do you find the number of positive roots of a polynomial?

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To find the number of positive roots of a polynomial, count the number of sign changes in the polynomial when evaluated at f(x).

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What is a zero of a polynomial?

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A zero of a polynomial is a value of x for which the polynomial evaluates to zero, indicating that (x - r) is a factor of the polynomial.

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

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The degree of a polynomial is equal to the total number of its zeros, including real and complex zeros, counted with their multiplicities.

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How many zeros does the polynomial f(x) = x^5 - 3x^3 + x have?

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The polynomial f(x) = x^5 - 3x^3 + x has 5 zeros.