From Wikipedia

Descartes' rule of signs

Positive roots

"The rule states that 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 is less than it by an even number. Multiple roots of the same value are counted separately."

Negative roots

As a corollary of the rule, the number of negative roots is the number of sign changes after multiplying the coefficients of odd-power terms by −1, or fewer than it by an even number. This procedure is equivalent to substituting the negation of the variable for the variable itself.

My question

I'm sorry for this question. I think is more an English one than a mathematical matter. I didn't understand the meaning of the phrases: "...less than it by an even number" and "fewer than it by an even number"

So if the change signs are $5$ or $6$ which "even number" he meant?

  • $\begingroup$ # of sign changes$-k$ for some even number $k$ $\endgroup$ – Bumblebee Nov 7 '17 at 3:19
  • $\begingroup$ @Bumblebee so the upper bound must be "# of sign changes $−k$ for some even number $k$"? $\endgroup$ – user42912 Nov 7 '17 at 3:23
  • $\begingroup$ @Bumblebee so if 6 is the number of sigh changes, we may have 4, 2 or 0 as upper bound?, so in this case it may have 3 positive roots, if the upper bound is 4? $\endgroup$ – user42912 Nov 7 '17 at 3:26
  • $\begingroup$ What do you mean by the upper bound? $\endgroup$ – Bumblebee Nov 7 '17 at 3:27
  • $\begingroup$ If the number of sign changes is $5$, then there are $1,3,$ or $5$ positive roots. $\endgroup$ – steven gregory Dec 30 '20 at 1:13

Let me explain by an example: Consider the polynomial $P(x)=x^7-5x^6+3x^4-x-2.$ This polynomial has $3$ sign changes. Therefore it has either $3$ or $1$ positive real root. Also $P(-x)=-x^7-5x^6+3x^4+x-2$ has two sign changes. Thus $P$ has two negative real roots or it has no negative real roots.

  • $\begingroup$ This conforms our observation $\endgroup$ – Bumblebee Nov 7 '17 at 3:37
  • $\begingroup$ Thank you for your answer. So It says "less than it by an even number" so can we interpret the word "less" as $-$? I thought "less" meant in this context as "<" $\endgroup$ – user42912 Nov 7 '17 at 3:38
  • $\begingroup$ I think so. I am not a native English speaker. but I have seen that people use this phrase when they talk about Descartes' rule of signs. $\endgroup$ – Bumblebee Nov 7 '17 at 3:41

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