The 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 function of degree n has exactly n roots in the complex number system, counting multiplicities.

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What is the degree of a polynomial?

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

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How do you determine the degree of the polynomial f(x) = x^3(x + 3)^2(x - 5)?

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Expand the polynomial: f(x) = x^3(x^2 + 6x + 9)(x - 5). The highest power is 6, so the degree is 6.

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What are the zeros of a polynomial?

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The zeros of a polynomial are the values of x for which the polynomial equals zero; they are also known as the roots.

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What does it mean for a polynomial to cross the x-axis?

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When a polynomial crosses the x-axis, it means that at that point, the value of the polynomial is zero, indicating a root.

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What is a leading coefficient?

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The leading coefficient is the coefficient of the term with the highest degree in a polynomial.

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Find the leading coefficient of f(x) = 4x^2 + 6 - 2x^3 + 5x^4.

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The leading coefficient is 5, from the term 5x^4.

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

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The maximum number of positive roots a polynomial can have is equal to its degree.

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

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Use Descartes' Rule of Signs: count the number of sign changes in the polynomial's coefficients.

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What is Descartes' Rule of Signs?

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Descartes' Rule of Signs states that the number of positive real roots of a polynomial is equal to the number of sign changes between consecutive non-zero coefficients.