2. Find the least common multiple of 2a bc, 5a2c3, 10ab3d, and 15abcd. Ans. 30a b'c'd.

3. Find the least common multiple of 3xy, 15xy3, 10xyz3, and 5x3y3z. Ans. 30x'y'z'.

4. Find the least common multiple of x+xy, xy-y3, and x2-y2. Ans. x'y-xy'.

5. Find the least common multiple of x-a1, x2—a2, x2+a3, and x-2a2x2+a*. Ans. x-a1x-a*x2+a®. 6. Find the least common multiple of x3-x, x3--1, and ∞3+1. x2-4 7. Find the least common multiple of x*+2x2+1, x*—2x2+1, x2+2x+1, x3—2x+1, x+1, and x-1. Ans. x-2x+1.

8. What is the least common multiple of 4x3+2x, 6x2—4x, and 6x2+4x?

Ans. 36x+2x3—8x.

4

9. What is the least common multiple of x2-4a3, (x+2a)3, and (x-2a)? x2 - 12x " a2 + 48 x2a"-64a6 10. What is the least common multiple of a-bʻ, a'—b3, a'—b3, and a-b? Ans. a®+ab+a*b3—a*b*—al3—bo.

CASE II.

110. When the quantities can not be factored by inspection.

The rule for this case may be deduced as follows:

1-If two polynomials are prime to each other, their product must be their least common multiple.

2. If two polynomials have a common divisor, their product must contain the second power of this common divisor; their least common multiple will therefore be obtained, by suppressing the first power of the common divisor in the product, or in one of the given quantities before multiplication.

3.—If we find the least common multiple of two polynomials, and then the least common multiple of this result and a third polynomial, and so on, the last result will evidently contain all the factors of the given polynomials, and no other factors. It will, therefore, be the least common multiple of the polynomials (109, 3).

Hence the following

RULE. I. When only two polynomials are given :

Find the greatest common divisor of the given polynomials; suppress this divisor in one of the polynomials, and multiply the result by the other polynomial.

II. When three or more polynomials are given :—

Find the least common multiple of any two of the polynomials; then find the least common multiple of this result and a third polynomial; and so on, till all the polynomials have been used. The last result will be the least common multiple required.

NOTE. It will generally be found preferable to commence with the greatest and next greatest of the given quantities.

EXAMPLES FOR PRACTICE.

Find the least common multiple

1. Of x2+x2-4x+6 and x3-5x+8x-6.

Ans. x-2x-7x+18x-18.

2. Of x3-2x2-19x+20 and x2-12x+35.

Ans. x-9x-5x+153x-140.

3. Of 6a'm'-am-1 and 2a'm*+3am1-2.

Ans. 6a'm'+11am-3am-2.

4. Of 2x-5x-x+1 and x-5x+7x-2.

Ans. 2x-9x+9x+3x-2.

5. Of 3x+6x-5x-10 and 6x-4x2-10.

Ans. 6x+12x-4x3-8x-10x-20.

6. Of x+7x+10, x2-2x-8, and x+x-20.

Ans. x+3x-18x-40.

7. Of a-3ab+2b', a'-ab-26', and a2-b'.

Ans. a'-2a'b-a b2+2b3.

8. Of 2x2-7xy+3y', 2x-5xy+2y, and x2-5xy+6y'.

Ans. 2x-11xy+17xy3—6y3.

FRACTIONS.

DEFINITIONS AND NOTATION.

111. We have seen (12) that division may be indicated by writing the dividend and divisor on opposite sides of a horizontal line. The term Fraction, in Algebra, relates to this mode or form of indicating division. Hence,

α

112. A Fraction is a quotient expressed by writing the dividend above a horizontal line, and the divisor below. Thus is a fracb tion, and is read, a divided by b.

113. The Denominator of the fraction is the quantity below the line, or the divisor.

114. The Numerator is the quantity above the line, or the dividend.

115. Any fraction may be decomposed as follows:

1. The value of a fraction is equal to the reciprocal of the denominator multiplied by the numerator.

2. In any fraction, the reciprocal of the denominator may be regarded as a fractional unit; and the numerator shows how many times this unit is taken in the fraction. Hence,

3.—A fraction is a fractional unit or a collection of fractional units, the value of each depending upon the denominator.

116. An Entire Quantity is an algebraic expression which has no fractional part; as x3-3xy.

117. A Mixed Quantity is one which has both entire and frac tional parts; as a2 +

GENERAL PRINCIPLES OF FRACTIONS.

118. Since a fraction is a form of expressing division, it is evident that all the operations in fractions must be based upon the general relations subsisting between the dividend, divisor, and quotient. These principles relate, first, to change of value; second, to change of sign.

1st. Change of value.

119. By modifying the language of (84), we may express the mutual relations of the numerator and denominator of a fraction, as follows:

I. Multiplying the numerator multiplies the fraction, and dividing the numerator divides the fraction.

II. Multiplying the denominator divides the fraction, and dividing the denominator multiplies the fraction.

III. Multiplying or dividing both numerator and denominator by the same quantity, does not alter the value of the fraction.

2d. Change of sign.

120. The Apparent Sign of a fraction is the sign written before the dividing line, to indicate whether the fraction is to be added or subtracted. Thus, in

the apparent sign of the fraction is plus, and indicates that the fraction is to be added.

121. The Real Sign of a fraction is the sign of its numerical value, when reduced to a monomial, and shows whether the fraction is essentially a positive or a negative quantity. Thus, in the fraction just given, let a=2 and x=3. Then

The real sign of this fraction therefore is minus, though its apparent sign is plus.

122. Each term in the numerator and denominator of a fraction has its own particular sign, distinct from the real or apparent sign

of the fraction. Now the essential sign of any entire quantity is changed, by changing the signs of all its terms. Hence,

I. Changing all the signs of either numerator or denominator, changes the real sign of the fraction; (85, I).

II. Changing all the signs of both numerator and denominator, does not alter the real sign of the fraction; (85, II).

III. Changing the apparent sign of the fraction, changes the real sign.

REDUCTION.

123. The Reduction of a fraction is the operation of changing its form without altering its value.

CASE I.

124. To reduce a fraction to its lowest terms.

A fraction is in its lowest terms, when the numerator and denominator are prime to each other. And since it does not alter the value of a fraction to suppress the same factor in both numerator and denominator, (119, III), we have the following

RULE. I. Resolve the numerator and denominator into their prime factors, and cancel all those factors which are common Or,

II. Divide both numerator and denominator by their greatest common divisor.