Question

Use Descartes' Rule of Signs to analyze the zeros of the following. List all possibilities.

P(x) = x5 - 4x4 + 3x3 + 2x - 6

P(x) = -5x4 + x3 + 2x2 - 1

P(x) = 2x5 + 7x3 + 6x2 - 2

Answer 1

step1, count the number of sign changes
this is the possible number of positive roots

step2, change the odd degree signs and recount
this is the number of possible negative roots

step3, complex roots always come in pairs so we would have to adjust steps 1 and 2 to determine these

Answer 2

x5 - 4x4 + 3x3 + 2x - 6
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3 possible positive roots, pull out 2 for complex roots


-x5 - 4x4 - 3x3 - 2x - 6

0 possible negative roots