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Algebra
 Expansion of a Fraction
  Example
 Sources and References

Algebra

Expansion of a Fraction

A fractional expression such as 4π‘₯βˆ’10π‘₯21βˆ’6π‘₯+11π‘₯2βˆ’6π‘₯3 may be expanded in ascending powers of π‘₯ in three different ways.
  1. First, by dividing the numerator by the denominator in the ordinary way, or by Synthetic Division, as shewn in (28).
  2. Secondly, by the method of Indeterminate Coefficients (232).
  3. Thirdly, by Partial Fractions and the Binomial Theorem.

Example

To expand by the method of Indeterminate Coefficients, proceed as follows:- Assume 4π‘₯βˆ’10π‘₯21βˆ’6π‘₯+11π‘₯2βˆ’6π‘₯3=𝐴+𝐡π‘₯+𝐢π‘₯2+𝐷π‘₯3+𝐸π‘₯4+𝐹π‘₯5+β‹― 4π‘₯βˆ’10π‘₯2=𝐴+𝐡π‘₯+𝐢π‘₯2+𝐷π‘₯3+𝐸π‘₯4+𝐹π‘₯5+β‹―  βˆ’6𝐴π‘₯βˆ’6𝐡π‘₯2βˆ’6𝐢π‘₯3βˆ’6𝐷π‘₯4βˆ’6𝐸π‘₯5βˆ’β‹―     π΄π‘₯2+𝐡π‘₯3+𝐢π‘₯4+𝐷π‘₯5+β‹―      βˆ’6𝐴π‘₯3βˆ’6𝐡π‘₯4βˆ’6𝐢π‘₯5βˆ’β‹― Equate coefficients of like powers of π‘₯, thus       π΄=0,     π΅βˆ’6𝐴=4, βˆ΄ π΅=4;   πΆβˆ’6𝐡+11𝐴=βˆ’10, βˆ΄ πΆ=14; π·βˆ’6𝐢+11π΅βˆ’6𝐴=0, βˆ΄ π·=40; πΈβˆ’6𝐷+11πΆβˆ’6𝐡=0, βˆ΄ πΈ=110; πΉβˆ’6𝐸+11π·βˆ’6𝐢=0, βˆ΄ πΉ=304;  β‹― β‹― β‹― β‹―β‹―β‹―β‹― The formation of the same coefficients by synthetic division is now exhibited, in order that the connexion between the two processes may be clearly seen. The division of 4π‘₯βˆ’10π‘₯2 by 1βˆ’6π‘₯+11π‘₯2βˆ’6π‘₯3 is as follows:    | 0+4βˆ’10  +6 |     24+84+240+660  βˆ’11 |      βˆ’44βˆ’154βˆ’440βˆ’1210 +6 |        +24+84+240+660    | 0+4+14+40+110+304+β‹―β‹―β‹―    | π΄ π΅ πΆ π· πΈ πΉ     If we stop at the term 110π‘₯4, then the undivided remainder will be 304π‘₯5βˆ’970π‘₯6+660π‘₯7, and the complete result will be 4π‘₯+14π‘₯2+40π‘₯3+110π‘₯4+304π‘₯5βˆ’970π‘₯6+660π‘₯71βˆ’6π‘₯+11π‘₯2βˆ’6π‘₯3 248 Here the concluding fraction may be regarded as the sum to infinity after four terms of the series, just as the original expression is considered to be the sum to infinity of the whole series.249 If the general term be required, the method of expansion by partial fractions must be adopted. See (257), where the general term of the foregoing series is obtained.250

Sources and References

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

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ID: 210600022 Last Updated: 6/22/2021 Revision: 0 Ref:

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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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