Here the division may be carried on to any number of terms in
the quotient, and we can stop at any term we please by taking for our
remainder the fraction whose numerator is the remainder last found, and whose denominator is the divisor.
Thus, if we carried on the quotient to four terms, we should have
The terms in the quotient may be fractional ; thus if x^2 is divided by x^3— a^3, the first four terms of the quotient are {1 }{x} +{ a^3 }{x^4} + {a^6 }{x^7} + {a^9 }{x^10} }, and the remainder is {a^12}{x10}.
161. Miscellaneous examples in multiplication and division occur which can be dealt with by the preceding rules for the reduction of fractions.
Ex. Multiply x--2a — — by 2 x - a ^ ^^
The product =(x+2a - {a^2}{2x + 3a}) \times (2x - a - { 2a^2}{x + a} ={ 2 x^2 + 7 ax + 6 a^2 - a^2} {2x + 3a} {2x^2 + ax - a^2 - 2 a^2}{x + a } ={ 2 x^2 + 7 ax + 5 a^2}{ 2 x + 3 a} {2x^2 + ax - 3 a^2 }{x + a } = {(2 x+ 5 a) (x + a)}{2x + 3a} {(2x + 3a)(x- a)}{ x + a } = (2x + 5 a) (x — a).
EXAMPLES XIV. b.
Express each of the following fractions as a group of simple fractions in lowest terms:
^ 3 x^y + xxp' - if- ^ g + ?> + c Ox?/ ' abc 2 3 g^x — 4 a'^x^ + G ax^ g he + ca + ab 12 ax abc b-hS ab- + b^ g a^bc - 3 2a6 ' 6 abc 3 g3 - 3 g2?) + 3 g6- + 6=^ g gS^c - 3 abh- + 2 g6c
