Theory of Equation

Equal roots of an equation

By expanding 𝑓(𝑥+𝑧) in powers of 𝑧 by (405), and also by (426), and equating the coefficients of 𝑧 in the two expansions, it is proved that 432 𝑓'(𝑥)=𝑓(𝑥)(𝑥−𝑎)+𝑓(𝑥)(𝑥−𝑏)+𝑓(𝑥)(𝑥−𝑐)+⋯ from which result it appears that, if the roots 𝑎, 𝑏, 𝑐, ⋯ are all unequal, 𝑓(𝑥) and 𝑓'(𝑥) can have no common measure involving 𝑥. If, however, there are 𝑟 roots each equal to 𝑎, 𝑠 roots equal to 𝑏, 𝑡 roots equal to 𝑐, ⋯, so that 𝑓(𝑥)=𝑝₀(𝑥−𝑎)^𝑟(𝑥−𝑏)^𝑠(𝑥−𝑐)^𝑡⋯ then 433 𝑓'(𝑥)=𝑟𝑓(𝑥)𝑥−𝑎+𝑠𝑓(𝑥)𝑥−𝑏+𝑡𝑓(𝑥)𝑥−𝑐+⋯ and the greatest common measure of 𝑓(𝑥) and 𝑓'(𝑥) will be 444 (𝑥−𝑎)^𝑟−1(𝑥−𝑏)^𝑠−1(𝑥−𝑐)^𝑡−1⋯ When 𝑥=𝑎, 𝑓(𝑥), 𝑓'(𝑥), ⋯, 𝑓^𝑟−1(𝑥) all vanish. Similarly when x=𝑏, ⋯.

Practical method of finding the equal roots

445 Let 𝑓(𝑥)=𝑋₁𝑋²₂𝑋³₃𝑋⁴₄𝑋⁵₅⋯𝑋⁴₄𝑋^𝑚_𝑚 where 𝑋₁≡product of all the factors like (𝑥−𝑎) 𝑋²₂≡product of all the factors like (𝑥−𝑎)² 𝑋³₃≡product of all the factors like (𝑥−𝑎)³ Find the greatest common measure of 𝑓(𝑥) and 𝑓'(𝑥)=𝐹₁(𝑥) say, the greatest common measure of 𝐹₁(𝑥) and 𝐹'₁(𝑥)=𝐹₂(𝑥) the greatest common measure of 𝐹₂(𝑥) and 𝐹'₂(𝑥)=𝐹₃(𝑥) ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ Lastly, the greatest common measure of 𝐹_𝑚−1(𝑥) and 𝐹'_𝑚−1(𝑥)=𝐹_𝑚(𝑥)=1 Next perform the divisions 𝑓(𝑥)÷𝐹₁(𝑥)=𝜙₁(𝑥) say 𝐹₁(𝑥)÷𝐹₂(𝑥)=𝜙₂(𝑥), ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ 𝐹_𝑚−1(𝑥)÷1=𝜙_𝑚(𝑥), And, finally, 𝜙₁(𝑥)÷𝜙₂(𝑥)=𝑋₁, 𝜙₂(𝑥)÷𝜙₃(𝑥)=𝑋₂, ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ 𝜙_𝑚−1(𝑥)÷𝜙_𝑚(𝑥)=𝑋_𝑚−1, 𝐹_𝑚−1(𝑥)=𝜙_𝑚(𝑥)=𝑋_𝑚T. 82 The solution of the equations 𝑋₁=0, 𝑋₂=0, ⋯ will furnish all the roots of 𝑓(𝑥); those which occure twice being found from 𝑋₂; those which occur three times each, from 𝑋₃, ⋯. 446 If 𝑓(𝑥) has all its coefficients commensurable, 𝑋₁, 𝑋₂, 𝑋₃, ⋯ have likewise their coefficients commensurable. Hence, if only one root be repeated 𝑟 times, that root must be commensurable. 447 In all the following theorems, unless otherwise stated, 𝑓(𝑥) is understood to have unity for the coefficient of its first term.

Sources and References

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