Question

Using complete sentences, describe how you would analyze the zeros of the polynomial function f(x) = 7x5 + 15x4 – x3 + 4x2 – 6x – 11 using Descartes’ Rule of Signs.

Answer 1

Positive roots
look at the polynomial as it stands f(x)
then the number of positive roots of the polynomial is either equal to the number of sign differences between consecutive nonzero coefficients, or is less than it by a multiple of 2.

f(x) = 7x^5 + 15x^4 – x^3 + 4x^2 – 6x – 11
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as u can see there r 3 sign differences so There are 3 or 1 positive roots. maximum possible number of positive roots : 3


Negative roots
we form f(-x)
f(-x) = -7x^5 + 15x^4 + x^3 + 4x^2 + 6x – 11
then the number of negative roots of the polynomial is either equal to the number of sign differences between consecutive nonzero coefficients of f(-x), or is less than it by a multiple of 2.
f(-x) = -7x^5 + 15x^4 + x^3 + 4x^2 + 6x – 11
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as u can see there r 2 sign differences so There are 2 or 0 negative roots. maximum possible number of negative roots : 2