Algebra

Expansion of a Fraction

A fractional expression such as 4𝑥−10𝑥²1−6𝑥+11𝑥²−6𝑥³ may be expanded in ascending powers of 𝑥 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 4𝑥−10𝑥21−6𝑥+11𝑥2−6𝑥3=𝐴+𝐵𝑥+𝐶𝑥2+𝐷𝑥3+𝐸𝑥4+𝐹𝑥5+⋯ 4𝑥−10𝑥2=𝐴+𝐵𝑥+𝐶𝑥2+𝐷𝑥3+𝐸𝑥4+𝐹𝑥5+⋯  −6𝐴𝑥−6𝐵𝑥2−6𝐶𝑥3−6𝐷𝑥4−6𝐸𝑥5−⋯     𝐴𝑥2+𝐵𝑥3+𝐶𝑥4+𝐷𝑥5+⋯      −6𝐴𝑥3−6𝐵𝑥4−6𝐶𝑥5−⋯ Equate coefficients of like powers of 𝑥, thus       𝐴=0,     𝐵−6𝐴=4, ∴ 𝐵=4;   𝐶−6𝐵+11𝐴=−10, ∴ 𝐶=14; 𝐷−6𝐶+11𝐵−6𝐴=0, ∴ 𝐷=40; 𝐸−6𝐷+11𝐶−6𝐵=0, ∴ 𝐸=110; 𝐹−6𝐸+11𝐷−6𝐶=0, ∴ 𝐹=304;  ⋯ ⋯ ⋯ ⋯⋯⋯⋯ 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 4𝑥−10𝑥² by 1−6𝑥+11𝑥²−6𝑥³ is as follows:    | 0+4−10  +6 |     24+84+240+660  −11 |      −44−154−440−1210 +6 |        +24+84+240+660    | 0+4+14+40+110+304+⋯⋯⋯    | 𝐴 𝐵 𝐶 𝐷 𝐸 𝐹     If we stop at the term 110𝑥⁴, then the undivided remainder will be 304𝑥⁵−970𝑥⁶+660𝑥⁷, and the complete result will be 4𝑥+14𝑥²+40𝑥³+110𝑥⁴+304𝑥⁵−970𝑥⁶+660𝑥⁷1−6𝑥+11𝑥²−6𝑥³ 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