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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 is a polynomial?

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A polynomial is a mathematical expression consisting of variables (or indeterminates) raised to non-negative integer powers and coefficients, combined using addition, subtraction, and multiplication.

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What is a root of a polynomial?

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A root of a polynomial is a value of x that makes the polynomial equal to zero.

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How do you determine the degree of a polynomial?

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The degree of a polynomial is determined by the highest power of the variable in the expression.

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What is the relationship between the degree of a polynomial and the number of roots?

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The degree of a polynomial indicates the maximum number of roots it can have, including complex and repeated roots.

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What is a rational root?

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A rational root is a solution to a polynomial equation that can be expressed as a fraction of two integers.

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What is Descartes' Rule of Signs?

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Descartes' Rule of Signs provides a way to determine 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 can you find possible rational roots of a polynomial?

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Possible rational roots can be found using the Rational Root Theorem, which states that any rational solution, expressed as p/q, has p as a factor of the constant term and q as a factor of the leading coefficient.

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What does it mean for a polynomial to have multiple roots?

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A polynomial has multiple roots when a particular root occurs more than once, which affects the behavior of the polynomial at that root.

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What is the significance of complex roots in polynomials?

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Complex roots occur in conjugate pairs for polynomials with real coefficients, and they contribute to the total count of roots as stated by the Fundamental Theorem of Algebra.