Polynomial Inequalities

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What is a polynomial inequality?

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A polynomial inequality is an inequality that involves a polynomial expression, typically in the form f(x) < 0, f(x) > 0, f(x) ≤ 0, or f(x) ≥ 0.

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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 roots or solutions.

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What does it mean for a zero to have multiplicity?

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Multiplicity refers to the number of times a particular zero appears as a root of the polynomial. For example, if (x - 1)² is a factor, then 1 is a zero with multiplicity 2.

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How do you determine where a polynomial is less than zero?

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To determine where a polynomial is less than zero, find the zeros, test intervals between the zeros, and check the sign of the polynomial in those intervals.

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What is Descartes's rule of signs?

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Descartes's rule of signs is a method used to determine the number of positive and negative real zeros of a polynomial based on the number of sign changes in the polynomial's coefficients.

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How do you apply Descartes's rule of signs for positive zeros?

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Count the number of sign changes in f(x) to determine the maximum number of positive real zeros. The actual number of positive zeros is either equal to this count or less by an even number.

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How do you apply Descartes's rule of signs for negative zeros?

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To find negative zeros, evaluate f(-x) and count the sign changes in the resulting polynomial to determine the maximum number of negative real zeros.

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What is the significance of the interval notation in polynomial inequalities?

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Interval notation is used to represent the solution set of polynomial inequalities, indicating the ranges of x-values that satisfy the inequality.

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What does it mean for a polynomial to be increasing or decreasing?

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A polynomial is increasing on an interval if its derivative is positive in that interval, and decreasing if its derivative is negative.

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

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The degree of a polynomial indicates the maximum number of real zeros it can have, according to the Fundamental Theorem of Algebra.