Theory of Equation

Descartes' Rule of Signs

416 In the following theorems every two adjacent terms in 𝑓(𝑥), which have the same signs, count as one "continuation of sign"; and every two adjacent terms, with different signs, count as one change of sign. 417 𝑓(𝑥), multiplied by (𝑥−𝑎), has an odd number of changes of sign thereby introduced, and one at least. 418 𝑓(𝑥) cannot have more positive roots than changes of sign, or more negative roots than continuations of sign. 419 When all the roots of 𝑓(𝑥) are real, the number of positive roots is equal to the number of changes of sign in 𝑓(𝑥); and the number of negative roots is equal to the number of changes of sign in 𝑓(−𝑥). 420 Thus, it being known that the roots of the equation 𝑥⁴−10𝑥³+35𝑥²−50𝑥+24=0 are all real; the number of positive roots will be equal to the number of changes of sign, which is four. Also 𝑓(−𝑥)=𝑥⁴+10𝑥³+35𝑥²+50𝑥+24=0, and since there is no change of sign, there is consequently, by the rule, no negative root. 421 If the degree of 𝑓(𝑥) exceeds the number of changes of sign in 𝑓(𝑥) and 𝑓(−𝑥) together, by 𝜇, there are at least 𝜇 imaginary roots. 422 If, between two terms in 𝑓(𝑥) of the same sign, there be an odd number of consecutive terms wanting, then there must be at least one more than that number of imaginary roots; and if the missing terms lie between terms of different sign, there is at elast one less than the same number of imaginary roots. Thus, in the cubic 𝑥³+4𝑥−7=0< There must be two imaginary roots. And in the equation 𝑥⁶−1=0< there are, for certain, four imaginary roots. 423 If an even number of consecutive terms be wanting in 𝑓(𝑥), there is at least the same number of imaginary roots. Thus the equation 𝑥⁵+1=0 has four terms absent; and therefore four imaginary roots at least.

Sources and References

https://archive.org/details/synopsis-of-elementary-results-in-pure-and-applied-mathematics-pdfdrive