75 a. When the given quantities cannot readily be factored, another method is used.

The L. C. M. of two polynomials contains all the factors of each polynomial; the G. C. D. contains all the factors common to the two polynomials. Hence, to find the L. C. M. of two polynomials, we have the following rule:

Find the G. C. D. of the two quantities. Divide one of the quantities by it, then multiply the other quantity by the .quotient.

Exercises.

Find the L. C. M. of the following:1. 2x2-xy-6y' and 3x-8xy+4y2. Their G. C. D. is x- 2y hence their L. C. M. is

2x2-xy-6y2x (3x2 - 8 xy +4y2) = (2x + 3y)(3x2 - 8 xy + 4y3);

x-2y

2. 3x-5x+2 and 423-4x2-x+1.

To find the L. C. M. of several quantities,

Find the L. C. M. of the first and second, then of that result and the third, and so on to the last.

Find the L. C. M. of the following:

4. x-6x+11x-6, x3-9x2+26x-24,

and x3-8x+19x - 12.

Ans. (x-1)(x 2)(x − 3)(x — 4).

5. 210x+9, x+10x+20x10x-21,

and x+4x-22x-4x+21.

Ans. (x2 — 1)(x2 — 9)(x+7).

6. x2 - (a+b)x + ab, x2 − (b+c)x+bc,

7. 6(a3 — b3)(a — b)3, 9 (a1 — b1) (a — b)2, and 12(a3 — b2)3. Ans. 36 (a*b*)(a3 — b3)(a2 — b2)2.

8. x2+5x+6, x-2x-8, and x-x-12.

Ans. (x+3)(x-4)(x+2).

9. 6x13x+6, 6x2-5x-6, and 4x2-9.

Ans. (3x+2)(2x+3)(2x − 3).

10. 2x2+11x+15, 2x2 + x − 10, and x2 + x − 6.

CHAPTER IV.

FRACTIONS.

76. A fractional unit is any one of a number of equal parts

77. A fraction is a fractional unit, or a collection of 1 3 5 a fractional units. Thus, 2 4' 7' b

are fractions.

78. Every fraction is composed of two parts, the denominator and the numerator. The denominator shows into how many equal parts the unit 1 is divided; and the numerator, how many of these parts are taken. Thus, in the fraction

b'

the denominator b shows that 1 is divided into b equal parts, and the numerator a shows that a of these parts are taken. The fractional unit, in all cases, is equal to the reciprocal of the denominator.

79. An entire quantity is one which contains no fractional part. Thus, 7, 11, a3x, 4x2 - 3y, are entire quantities. An entire quantity may be regarded as a fraction whose denominator is 1. Thus, 7

=

ab

ab

1

80. A mixed quantity is a quantity containing both entire and fractional parts. Thus, 7, 84, a+b

are mixed

quantities.

81. Let denote any fraction, and q any quantity what

b

ever. From the preceding definitions,

α

denotes that is

170

taken a times; also a denotes that is taken aq times,

Multiplying the numerator of a fraction by any quantity is equivalent to multiplying the fraction by that quantity. We see, also, that

Any quantity may be multiplied by a fraction by multiplying it by the numerator, and then dividing the result by the denominator.

82. It is a principle of division that the same result will be obtained if we divide the quantity a by the product of two factors, p× q, as would be obtained by dividing it first by one of the factors, p, and then dividing that result by the other factor, q; that is,

Multiplying the denominator of a fraction by any quantity is equivalent to dividing the fraction by that quantity.

83. Since the operations of multiplication and division are the converse of each other, it follows, from the preceding principles, that

Dividing the numerator of a fraction by any quantity is equivalent to dividing the fraction by that quantity.

Dividing the denominator of a fraction by any quantity is equivalent to multiplying the fraction by that quantity.

84. Since a quantity may be multiplied and the result divided by the same quantity without altering the value, it follows that

Both terms of a fraction may be multiplied by any quantity, or both divided by any quantity, without changing the value of the fraction.

TRANSFORMATION OF FRACTIONS.

85. The transformation of a quantity is the operation of changing its form without altering its value. "reduce" has a technical signification, and means form."

The term "to trans

86. To reduce an Entire Quantity to a Fractional Form having a Given Denominator.

Let a be the quantity, and b the given denominator. We

have, evidently, a

=

ab

b

Hence the rule:

Multiply the quantity by the given denominator, and write the product over this given denominator.

87. To reduce a Fraction to its Lowest Terms.

A fraction is in its lowest terms when the numerator and the denominator contain no common factors.

It has been shown that both terms of a fraction may be divided by the same quantity without altering its value: therefore, if they have any common factors, we may strike them out. Hence the rule:

Resolve each term of the fraction into its prime factors, then strike out all that are common to both.

The same result is attained by dividing both terms of the fraction by any quantity that will divide them without a remainder, or by dividing them by their G. C. D.