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Theory of Equation
 Theory of Numbers
  Example
 Sources and References

Theory of Equation

Theory of Numbers

400 General form of a rational integral equation of the 𝑛th degree. 𝑝0π‘₯𝑛+𝑝1π‘₯π‘›βˆ’1+𝑝2π‘₯π‘›βˆ’2+β‹―+π‘π‘›βˆ’1π‘₯+𝑝𝑛 The left side will be designated 𝑓(π‘₯) in the following summary. 401 If 𝑓(π‘₯) be divided by π‘₯βˆ’π‘Ž, the remainder will be 𝑓(π‘Ž). By assuming 𝑓(π‘₯)=𝑃(π‘₯βˆ’π‘Ž)+𝑅. 402 If π‘Ž be a root of the equation 𝑓(π‘₯)=0, then 𝑓(π‘Ž)=0. 403 To compute 𝑓(π‘Ž) numerically; divide 𝑓(π‘₯) by π‘₯βˆ’π‘Ž, and the remainder will be 𝑓(π‘Ž). 404

Example

To find the value of 4π‘₯6βˆ’3π‘₯5+12π‘₯4βˆ’π‘₯2+10 when π‘₯=2.
 |4βˆ’3+12+ 0βˆ’ 1+  0+ 10
2|  8+10+44+88+174+348
----------------------
  4+5+22+44+87+174+358
Thus 𝑓(2)=358 405 If π‘Ž,𝑏,𝑐, β‹―, π‘˜ be the roots of the equation 𝑓(π‘₯)=0; then, by (401) and (402), 𝑓(π‘₯)=𝑝0(π‘₯βˆ’π‘Ž)(π‘₯βˆ’π‘)(π‘₯βˆ’π‘)β‹―(π‘₯βˆ’π‘˜) By multiplying out the last equation, and equating coefficients with equation (400), considering 𝑝0=1, the following results are obtained:- 406 βˆ’π‘1=the sum of all the roots of 𝑓(π‘₯). 𝑝2=the sum of the products of the roots taken two at a time. βˆ’π‘3=the sum of the products of the roots taken three at a time. β‹― (βˆ’1)π‘Ÿπ‘π‘Ÿ=the sum of the products of the roots taken three at π‘Ÿ a time. β‹― (βˆ’1)𝑛𝑝𝑛=product of all roots. 407 The number of roots of 𝑓(π‘₯) is equal to the degree of the equation. 408 Imaginary roots must occur in pairs of the form 𝛼+π›½βˆ’1, π›Όβˆ’π›½βˆ’1 The quadratic factor corresponding to these roots will then have real coefficients; for it will be π‘₯2βˆ’2𝛼π‘₯+𝛼2+𝛽2405, 226 409 If 𝑓(π‘₯) be of an odd degree, it has at least one real root of the opposite sign to 𝑝𝑛. Thus π‘₯2βˆ’1=0 has at least one positive root. 410 if 𝑓(π‘₯) be of an even degree, and 𝑝𝑛 negative, there is at least one positive and one negative root. Thus π‘₯4βˆ’1 has +1 and βˆ’1 for roots. 411 If several terms at the beginning of the equation are of one sign, and all the rest of another, there is one, and only one, positive root. Thus π‘₯5+2π‘₯4+3π‘₯3+π‘₯2βˆ’5π‘₯βˆ’4=0 has only one positive root. 412 If all the terms are positive there is no positive root. 413 If all the terms of an even order are of one sign, and all the rest are of another sign, there is no negative root. 414 Thus π‘₯4βˆ’π‘₯3+π‘₯2βˆ’π‘₯+1=0 has no negative root. 415 If all the indices are even, and all the terms of the same sign, there is no real root; and if all the indices are odd, and all the terms of the same sign, there is no real root but zero.
Thus π‘₯4+π‘₯2+1=0 has no real root, and π‘₯5+π‘₯3+π‘₯=0 has no real root but zero. In this last equation there is no absolute term, because such a term would involve the zero power of π‘₯, which is even, and by hypothesis is wanting.

Sources and References

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

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References

  1. B. Joseph, 1978, University Mathematics: A Textbook for Students of Science & Engineering
  2. Wheatstone, C., 1854, On the Formation of Powers from Arithmetical Progressions
  3. Stroud, K.A., 2001, Engineering Mathematics
  4. Coolidge, J.L., 1949, The Story of The Binomial Theorem
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