• anonymous
Describe how you would analyze the zeros of the polynomial function f(x) = -3x5 - 8x4 +25x3 - 8x2 +x - 19 using Descartes' Rule of Signs and be sure to include the answer
  • Stacey Warren - Expert
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  • katieb
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  • mathmate
-3x5 - 8x4 +25x3 - 8x2 +x - 19 Step 1: List the signs of each term --+-+- Note 4 changes in sign (from - to + or from + to -) So there is a maximum of 4 positive real roots (i.e. 0, 2 or 4) Step 2: Reverse the sign of odd-powers for the case of negative roots +----- So there is one change of sign, so there is a maximum of 1 real negative root. For complex roots, the minimum number is 5-(4+1)=0. Therefore, there can be 0, 2 or 4 complex roots, which is what will affect the number of positive or negative roots. In this case, it will only affect positive roots, since complex roots come in pairs. [note: (not derived from Descartes' rule of signs)] For this particular problem, there is only one real root, i.e. the negative root. The rest are all complex.
  • mathmate
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