113. Since the G.C.D. of two quantities contains all the factors common to both, if we divide the product of two quantities by their G. C.D., the quotient will be their L.C.M.

1. Find the L.C.M. of 6a2, 9ax3, and 24x3. Ans. 72a2x5. 32x2y2, 40ax3у, 5a2x(x—y). Ans. 160a2x3y2(x—y).

2.

5.

6.

7.

x—1, x2—1, x-2, and x2-4. Ans. x1-5x2+4.
x2—1, x2+1, (x−1)2, (x+1)2, î3—1, and x3+1.
Ans. x10-x-x1+1.

3x2-11x+6, 2x2—7x+3, and 6x2-7x+2. (See
Art. 113.)
Ans. 6x3-25x2+23x-6.

III. ALGEBRAIC FRACTIONS.

DEFINITIONS.

114. Algebraic Fractions are represented in the same manner as common fractions in arithmetic.

The quantity below the line is called the denominator, because it denominates, or shows the number of parts into which the unit is divided; the quantity above the line is called the numerator, because it numbers, or shows how many parts are taken.

a- -b Thus, in the fraction a unit is supposed to be dic+d' vided into cd equal parts, and a-b of those parts are taken.

115. The terms proper, improper, simple, compound, ard complex, have the same meaning when applied to algebraic fractions, as to common numerical fractions.

116. An Entire Algebraic Quantity is one not expressed under the form of a fraction.

117. A Mixed Quantity is one composed of an entire quantity and a fraction.

118. Proposition.-The value of a fraction is not altered, when both terms are multiplied or divided by the same quantity.

A

MA
mB

Let Q. Then, will -Q. For, since the numerator of a

fraction may always be considered a dividend, and the denominator a divisor, if we multiply the numerator or dividend by any quantity, as m, the quotient will be increased m times; if we multiply the denominator or divisor by m, the quotient will be diminished as much, or it will be divided by m. Therefore, the value of the fraction is not changed.

A similar method of reasoning may be applied to the division of the terms of a fraction.

Case I. TO REDUCE A FRACTION TO ITS LOWEST TERMS.

119. From Art. 118, we have the following

Rule.-Divide both terms of the fraction by any quantity that will exactly divide them, and continue this process as long as possible.

Or, Divide both terms by their greatest common divisor.

Or, Resolve both terms into their prime factors, and then cancel those factors which are common.

In algebraic fractions, the last is generally the best method.

The following examples are to be solved by factoring, but the process requires care and practice.

5x+1

Ans.

9x-4x

120. Exercises in Division, in which the quotient is a fraction, and capable of being reduced:

Case II.-TO REDUCE A FRACTION TO AN ENTIRE OR

MIXED QUANTITY.

121. Since the numerator of the fraction may be regarded as a dividend, and the denominator as the divisor, this is merely a case of division. Hence,

Rule.-Divide the numerator by the denominator, for the entire part. If there be a remainder, place it over the denominator, for the fractional part, and reduce it to its lowest

Case III. TO REDUCE A MIXED QUANTITY TO THE FORM

OF A FRACTION.

122. This is, obviously, the reverse of Case II. Hence, we have the following

Rule.-1. Multiply the entire part by the denominator of the fraction.

2. Add the numerator to the product, if the sign of the fraction be plus, or subtract it, if the sign be minus. 3. Place the result over the denominator.

Before applying this rule, it is necessary to consider

123. The Signs of Fractions.-Each of the several terms of the numerator and denominator of a fraction is preceded by the sign plus or minus, expressed or understood; and the fraction, taken as a whole, is also preceded by the sign plus or minus, expressed or understood.

Thus, in the fraction

a2_b2
x+y

the sign of a2 is plus; of b2, minus;

while the sign of each term of the denominator is plus; but the

sign of the fraction, taken as a whole, is minus.