Fundamental Theorem of Algebra

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What does the Fundamental Theorem of Algebra state?

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Every non-constant polynomial function of degree n has exactly n roots in the complex number system, counting multiplicities.

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How many roots does a polynomial of degree 3 have according to the Fundamental Theorem of Algebra?

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A polynomial of degree 3 has exactly 3 roots in the complex number system.

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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 for which the polynomial evaluates to zero.

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What is the Rational Root Theorem?

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The Rational Root Theorem states that any rational solution of a polynomial equation, in the form of p/q, is such that p is a factor of the constant term and q is a factor of the leading coefficient.

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If a polynomial has a degree of 5, how many zeros does it have?

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It has exactly 5 zeros, counting multiplicities.

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What are complex roots?

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Complex roots are solutions to polynomial equations that include imaginary numbers, typically in the form a + bi, where a and b are real numbers.

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What is the significance of multiplicity in polynomial roots?

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Multiplicity refers to the number of times a particular root is repeated in a polynomial. A root with a higher multiplicity indicates that the polynomial touches the x-axis at that root.

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How can you determine the number of real roots of a polynomial?

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You can determine the number of real roots by analyzing the polynomial's degree, its discriminant, and using techniques such as synthetic division or graphing.

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What is synthetic division and how is it used in finding roots?

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Synthetic division is a simplified form of polynomial long division used to divide a polynomial by a linear factor, helping to find roots more efficiently.

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What is the relationship between the coefficients of a polynomial and its roots?

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The coefficients of a polynomial are related to its roots through Vieta's formulas, which express the sums and products of the roots in terms of the coefficients.