Theory of Equation

Limits of the Roots

448 If the greatest negative coefficients in 𝑓(𝑥) and 𝑓(−𝑥) be 𝑝 and 𝑞 respectively, then 𝑝+1 and −(𝑞+1) are limits of the roots. 449 If 𝑥^𝑛−𝑟 and 𝑥^𝑛−𝑠 are the highest negative terms in 𝑓(𝑥) and 𝑓(−𝑥) respectively, (1+^𝑟𝑝) and −(1+^𝑠𝑞) are limits of the roots. 450 If 𝑘 be a superior limit to the positive roots of 𝑓1𝑥, then 1𝑘 will be an inferior limit to the positive roots of 𝑓(𝑥). 451 If each negative coefficient be divided by the sum of all the preceding positive coefficients, the greatest of the fractions so formed + unity will be a superior limit to the positive roots. 452

Newton's method

Put 𝑥=ℎ+𝑦 in 𝑓(𝑥); then, by (426), 𝑓(ℎ+𝑦)=𝑓(ℎ)+𝑦𝑓'(ℎ)+𝑦²|2𝑓²(ℎ)+⋯+𝑦^𝑛|𝑛𝑓^𝑛(ℎ)=0 Take ℎ so that 𝑓(ℎ), 𝑓'(ℎ), 𝑓²(ℎ), ⋯, 𝑓^𝑛(ℎ) are all positive; then ℎ is a superior limit to the positive roots. 453 According as 𝑓(𝑎) and 𝑓(𝑏) have the same or different signs, the number of roots intermediate between 𝑎 and 𝑏 is even or odd. 454

Rolle's Theorem

One real root of the equation 𝑓'(𝑥) lies between every two adjacent real roots of 𝑓(𝑥). 455 Cor. 1: 𝑓(𝑥) cannot have more than one root greater than the greatest root in 𝑓'(𝑥); or more than one less than the least root in 𝑓'(𝑥). 456 Cor. 2: If 𝑓(𝑥) has 𝑚 real roots, 𝑓^𝑟(𝑥) has at least 𝑚−𝑟 real roots. 457 Cor. 3: If 𝑓^𝑟(𝑥) has 𝜇 imaginary roots, 𝑓(𝑥) has also 𝜇 at least. 458 Cor. 4: If 𝛼, 𝛽, 𝛾, ⋯, 𝜅 be the roots of 𝑓'(𝑥); then the number of changes of sign in the series of terms 𝑓(∞), 𝑓(𝛼), 𝑓(𝛽), 𝑓(𝛾), ⋯, 𝑓(−∞) is equal to the number of roots of 𝑓(𝑥).

Sources and References

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