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4 rst then apply the rules of radicals applicable to

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Unformatted text preview: real zeros (counting multiplicity) is one of the numbers {P , P − 2, P − 4, . . . }. • If N denotes the number of variations of sign in the formula for f (−x), then the number of negative real zeros (counting multiplicity) is one of the numbers {N , N − 2, N − 4, . . . }. A couple of remarks about Descartes’ Rule of Signs are in order. First, Descartes’ Rule of Signs gives us an estimate to the number of real zeros, not the actual value of the zeros. Second, Descartes’ Rule of Signs counts multiplicities. This means that, for example, if one of the zeros has multiplicity 2, Descsartes’ Rule of Signs would count this as two zeros. Lastly, note that the number of positive or negative real zeros always starts with the number of sign changes and decreases by an even number. For example, if f (x) has 7 changes in sign, then, counting multplicities, f has either 7, 5, 3, or 1 positive real zero. This implies that the graph of y = f (x) crosses the positive x-axis at least once. If f (−x) results...
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