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117. The actual processes of multiplication and division can often be partially or wholly avoided by a skilful use of factors.

It should be observed that the formule which the student has seen exemplified in the preceding pages are just as useful in their converse as in their direct application. Thus the formula for resolving into factors the difference of two squares is equally useful as enabling us to write down at once the product of the sum and the difference of two quantities.

Example 1. Multiply 2a+3b-c by 2a-3b+c.
These expressions may be arranged thus:

2a+(3b-c) and 2a — (3b−c).

Hence the product={2a+(3b−c)} {2a — (3b−c)}

= (2a)2 — (3b—c)2
=4a2-(9b2-6bc+c2)

=4a2-9b2+6bc-c2.

[Art. 86.]

Example 2. Divide the product of 2x2+x-6, and 6x2-5x+1 by 3x2+5x-2.

Denoting the division by means of a fraction, the required quotient

=

(2x2+x−6) (6x2-5x+1)

3x2+5x-2

(2x-3)(x+2) (3x-1) (2x-1)
(3x-1)(x+2)

=(2x-3)(2x-1).

EXAMPLES XIV. b.

Find the product of

1. 2x-7y+3z and 2x+7y-3z.

2. 3x2-4xy+7y2 and 3x2+4xy+7y2.

3. 5x2+5xy-9y2 and 5x2-5xy — 9y2.

4. 7x2-8xy+3y2 and 7x2+8xy-3y2.

5. x3+2x2y+2xy2+y3 and x3 — 2x2y+2xy2 — y3.

6. (x+y)2+2(x+y)+4 and (x+y)2−2(x+y)+4.

7. Multiply the square of a+3b by a2-6ab+9b2.

8. Multiply (a−b)2+ 1⁄2 (b−c)2+ 1⁄2 (c− a)2 by a+b+c.

CHAPTER XV.

FRACTIONS.

118. THOSE portions of the present chapter given to general proofs of the rules employed in the treatment of fractions may be omitted by the student reading the subject for the first time.

119. DEFINITION. If a quantity x be divided into b equal parts, and a of these parts be taken, the result is called the fraction of x.

a

b

If x be the unit, the fraction of x is called simply

a

b

a

"the

fraction"; so that the fraction represents a equal parts, b of

b

which make up the unit.

REDUCTION OF FRACTIONS.

120. To prove that

By b

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we mean a equal parts, b of which make up the unit...(1);

But

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that is,

Conversely,

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=ma

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Hence we have the following rule:

RULE I. The value of a fraction is not altered if we multiply or divide the numerator and denominator by the same quantity.

An algebraical fraction may therefore be reduced to an equivalent fraction by dividing numerator and denominator by any common factor; if this factor be the highest common factor, the resulting fraction is said to be in its lowest terms.

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NOTE. The beginner should be careful not to begin cancelling until he has expressed both numerator and denominator in the most convenient form, by resolution into factors where necessary.

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121. When the factors of the numerator and denominator cannot be determined by inspection, the fraction may be reduced to its lowest terms by dividing both numerator and denominator by the highest common factor, which may be found by the rules given in Chap. XII.

Example. Reduce to lowest terms

3x3-13x2+23x-21
15x3- 38x2 - 2x+21*

First Method. The H.C.F. of numerator and denominator is 3x-7.

Dividing numerator and denominator by 3x-7, we obtain as respective quotients x2 - 2x+3 and 5x2 - x − 3.

Thus

=

x2-2x+3

3x3 - 13x2+23x - 21 (3x-7) (x2 - 2x+3)
15x3-38x2-2x+21 (3x − 7) (5x2 − x − 3) 5x2

=

-x- -3°

This is the simplest solution for the beginner; but in this and similar cases we may often effect the reduction without actually going through the process of finding the highest common factor.

Second Method. By Art. 97, the H.C.F. of numerator and denominator must be a factor of their sum 18x3-51x2+21x, that is, of 3x (3x-7) (2x-1). If there be a common divisor it must clearly be 3x-7; hence arranging numerator and denominator so as to shew 3x-7 as a factor,

the fraction

122.

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If either numerator or denominator can readily be resolved into factors we may use the following method.

Example. Reduce to lowest terms

x3+3x2 - 4x 7x3-18x2+6x+5°

The numerator=x(x2+3x-4)=x (x+4) (x − 1).

Of these factors the only one which can be a common divisor is x-1. Hence, arranging the denominator,

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MULTIPLICATION AND DIVISION OF FRACTIONS.

123. RULE II. To multiply a fraction by an integer: multiply the numerator by that integer; or, if the denominator be divisible by the integer, divide the denominator by it.

The rule may be proved as follows:

α

(1) / represents a equal parts, b of which make up the unit ;

ac

b

represents ac equal parts, b of which make up the unit; and the number of parts taken in the second fraction is c times the number taken in the first;

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