155. Multiply their numerators together for a new numerator, and their denominators together for a new denominator; reduce the resulting fraction to its lowest terms, and it will be the product of the fractions required.

It has been already observed, (Art. 119), that when a fraction is to be multiplied by a whole quantity, the numerator is multiplied by that quantity, and the denominator is retained:

10x
b

and -X5= ; or, which is the same,

2x
b

making an improper fraction of the integral quantity, and

a

C

ac

then proceeding according to the rule, we have X 1

1 b

Hence, if a fraction be multiplied by its denominator, the

manner, the result being the same, whether the numerator be multiplied by a whole quantity, or the denominator divided by it, the latter method is to be preferred, when the denominator is some multiple of the multiplier; Thus, let

ad

ad

be the fraction, and e the a.ultiplier; then -Xc=

bc

adc

bc

bc

ad

ad ; and -XC= b bc

Also, when the numerator of one of the fractions to be multiplied, and the denominator of the other, can be divided by some quantity which is common to each of them, the quotients may be used instead of the fractions themselves; thus,

Here, (3x+2)×8x=24x2+16x= numerator, and 4X7 28= denominator ;

3a

the product required.

-x2

7x2

by

a-x

Here, (a2 -x2) × 7x2=(a+x) × (α—x) × 7x2= numerator (Art. 106), and 3a × (a—x) = `denominator; see Ex.

15, (Art. 79).

Hence, the product is (a+x) × (a−x) × 7x2

3a X (a-x)

the numerator and denominator by a-x)

=(dividing

7x2(a+x)

3α

5

5a+x

5

3

3

Then, (5a+x) x (3a-x) = 15a2-2ax-x2= new numera

tor, and 5X3=15 denominator: Therefore,

156. But, when mixed quantities are to be multiplied together, it is sometimes more convenient to proceed, as in the multiplication of integral quantities, without reducing them to improper fractions.

Ex. 9. It is required to find the continual product of

3α 2x2

a+b

and

ax

Ex. 10. It is required to find the continued product of

a1-x4 a+y

and

a-y
a-x

Ans. a+x.

Ex. 11. It is required to find the continued product of a2 - x2 α2 -b2

and

α
ax-x2

a2-ab

Ans.

Ans. xx3x2—}x.

Ex. 12. Multiply x-x+1 by x—}x. x2

To divide one fractional quantity by another.

RULE.

157. Multiply the dividend by the reciprocal of the divisor, or which is the same, invert the divisor, and proceed, in every respect, as in multiplication of algebraic fractions; and the product thus found will be the quotient required.

When a fraction is to be divided by an integral quantity; the process is the reverse of that in multiplication; or, which is the same, multiply the denominator by the integral, (Art. 120), or divide the numerator by it. The latter mode is to be preferred, when the numerator is a multiple of the divisor.

hence

X

a+b 5a-5x'

3

3α-3x a+b Su-3x 3(a-x)
a+b 5α-5x 5a-5x 5(a-x) 5

is the quo

tient required.

Ex. 3. Divide

by a+b.

quotient required.

158. But it is, however, frequently more simple in practice to divide mixed quantities by one another, without reducing them to improper fractions, as in division of integral quantities, especially when the division would terminate. Ex. 5. Divide xxxx by x2-x. x2-x)x1—§x3+Ÿ x2—√x(x2-3x+1

* *

4

4

Ex. 12. Divide xa — 1-3x2+x2+ — x−2 by 3x −2.

x4

6

3

4

3 Ans.x3-x2+1.

§ VII. RESOLUTION OF ALGEBRAIC FRACTIONS OR QUOTIENTS

INTO INFINITE SERIES.

159. An infinite series is a continued rank, or progression of quantities, connected together by the signs + or ; and usually proceeds according to some regular, or determined law.

Thus, ++} + {+}+{;+, &c.

Or, -+-+-+, &c.

In the first of which, the several terms are the reciprocals of the odd numbers 1, 3, 5, 7, &c.; and in the latter the reciprocals of the even numbers, 2, 4, 6, 8, &c., with alternate signs.

160. We have already observed (Art. 96), that if the first or leading term of the remainder, in the division of algebraic quantities, be not divisible by the divisor, the operation might be considered as terminated; or, which is the same, that the integral part of the quotient has been obtained. And, it has also been remarked, (Art. 89), that the division of the remainder by the divisor can be only indicated, or expressed, by a fraction thus, for example, if we have to divide a by ■+1, we write for the quotient: This, however, does not prevent us from attempting the division according to the