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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Define a zero of a polynomial function.

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A zero of a polynomial function is a value of x for which the polynomial evaluates to zero, meaning it is an x-intercept of the graph.

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What does it mean for a root to have multiplicity?

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A root has multiplicity if it is a solution to the polynomial equation more than once. For example, if (x - 1)² is a factor, then 1 is a root with multiplicity 2.

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How can you find the rational zeros of a polynomial?

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To find the rational zeros of a polynomial, use 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 is the Rational Root Theorem?

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The Rational Root Theorem provides a way to find all possible rational roots of a polynomial equation by listing the factors of the constant term and the leading coefficient.

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What is the significance of the degree of a polynomial?

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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 (including multiplicities) the polynomial can have.

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What are complex roots?

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Complex roots are solutions to polynomial equations that include imaginary numbers. They occur in conjugate pairs when the coefficients of the polynomial are real.

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

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The number of real roots can be determined using the Descartes' Rule of Signs, which counts the number of sign changes in the polynomial's coefficients.

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What is a polynomial function?

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A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients, typically expressed in the form f(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0.

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

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The zeros of a polynomial correspond to the x-intercepts of its graph, where the graph crosses or touches the x-axis.