SOLUTION: For the polynomial P(x)=x^3-2x^2+6x-4. what does Descartes' Rules of Signs tell you about the number of possible positive real zeros? Your help would be appreciated, I am also w

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: For the polynomial P(x)=x^3-2x^2+6x-4. what does Descartes' Rules of Signs tell you about the number of possible positive real zeros? Your help would be appreciated, I am also w      Log On


   

Question 149090This question is from textbook
: For the polynomial P(x)=x^3-2x^2+6x-4. what does Descartes' Rules of Signs tell you about the number of possible positive real zeros?
Your help would be appreciated, I am also working on this and I am going to compare it to see if I am correct. Thank you.
This question is from textbook

Answer by solver91311(24713) About Me  (Show Source):
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If the terms of a single-variable polynomial with real coefficients are ordered by descending variable exponent, then the number of positive roots of the polynomial is either equal to the number of sign differences between consecutive nonzero coefficients, or less than it by a multiple of 2.

Your polynomial function is already ordered by descending variable exponent, so just count the sign changes.

First term is positive
2nd term is negative, so this is 1 sign change
3rd term is positive, so this is the 2nd sign change
4th term is negative, so this is the 3rd sign change

Three sign changes means that there are 3 positive real roots, or 1 positive real root (2 less). In fact, there are exactly 3 roots because this is a 3rd degree polynomial (Fundamental Theorem of Algebra). Since complex roots always appear as conjugate pairs, it is not possible to have 3 complex roots, at least one of the roots must be real and it must be positive because of Descarte's rule. It is also possible that you have 3 real roots, all of which must be positive because of Descarte's rule.

We can check the answer by using Descarte's rule to determine the possible number of negative roots. Change the sign on each of the ODD numbered exponent terms (which is the same thing as evaluating P(-x)), and then count the sign changes:

P(-x)=-x^3-2x^2-6x-4

1st term negative
2nd term negative (no change)
3rd term negative (still no change)
4th term negative (still no change)

Since there were no sign changes, the number of possible negative roots is 0. This makes sense since the test for positive roots accounted for all of the possible roots.

Hope this helps.