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
xaxis at least once. If f (−x) results...
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 Fall '13
 Wong
 Algebra, Trigonometry, Cartesian Coordinate System, The Land, The Waves, René Descartes, Euclidean geometry

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