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from the division of the given polynomial by this factor for the other factor.

1. Take, for example, the polynomial

ab + ac;

in which, it is plain, that a is a factor of both terms: hence ab + ac = a (b + c).

2. Take, for a second example, the polynomial

ab2c5ab3 + ab2c2.

It is plain that a and b2 are factors of all the terms: hence ab2c + 5ab3 + ab2c2 ab2 (c + 5b + c2).

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3. Take the polynomial 25a-30a3b + 15a2b2; it is evident that 5 and a2 are factors of each of the terms. We may, therefore, put the polynomial under the form

5a2 (5a2 6ab + 3b2).

4. Find the factors of 3a2b + 9a2c + 18a2xy.

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6. Find the factors of 24a2b2cx-30a8b5cy +36a7b8cd6abc. Ans. 6abc (4abx - 5a7b4c5y + 6aob1d + 1).

By the aid of the formulas of Art. 48, polynomials having certain forms may be resolved into their binomial factors.

1. Find the factors of a2 + 2ab+ b2.

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GENERAL EXAMPLES.

1. Find the factors of the polynomial 6a3b + 8a2b5 -2ab.

2. Find the factors of the polynomial 15abc2 12db6c2.

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3bc2+9a3b5c

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5. Find the factors of the polynomial n3 + 2n2 + n. n3 + 2n2 + n = n (n2 + 2n + 1)

First,

= n (n + 1) × (n + 1)

= n (n + 1)2.

6. Find the factors of the polynomial 5a2bc + 10ab2c + 15abc2.

Ans. 5abc (a+2b+3c).

7. Find the factors of the polynomial a2x x3.

Ans. x (a + x) (a — x).

60. Among the different principles of algebraic division, there is one remarkable for its applications. It is enunciated thus:

The difference of the same powers of any two quantities is exactly divisible by the difference of the quantities.

Let the quantities be represented by a and b; and let m de note any positive whole number. Then,

am-bm

will express the difference between the same powers of a

and it is to be proved that am If we begin the division of

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.d b,

bm is exactly divisible by

- b.

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Dividing am by a the quotient is am-1, by the rule for the exponents. The product of a -b by am-1 being subtracted from the dividend, the first remainder is am-1b-bm, which can be put under the form,

Now, if the factor

b (am−1 — bm−1).

(am-1 — bm-1)

of the remainder, be divisible by ab, b times (am-1 — bm-1), must be divisible by ab, and consequently am-bm must

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If the difference of the same powers of two quantities is exactly divisible by the difference of the quantities, then, the difference of the powers of a degree greater by 1 is also divisible by it.

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But by the rules for division, we know that a2 62 is divis ible by ab; hence, from what has just been proved, a3 — b3 b3 must be divisible by ab, and from this result we conclude b and so on indefinitely: hence

that a1 - 64 is divisible by a the proposition is proved.

61. To determine the form of the quotient. If we continue the operation for division, we shall find am-26 for the second term of the quotient, and am-2f2 - bm for the second remainder; also, am-362 for the third term of the quotient, and am-363 — fm for the third remainder; and so on to the mth term of the quo tient, which will be

am-mfm-1 or bm-1;

and the mth remainder will be

ат-тът

bm or bm'bm = 0.

Since the operation ceases when the remainder becomes 0, we shall have m terms in the quotient, and the result may be written thus:

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CHAPTER III.

OF ALGEBRAIC FRACTIONS.

62. AN ALGEBRAIC FRACTION is an expression of one or more equal parts of 1.

One of these equal parts is called the fractional unit. Thus, a is an algebraic fraction, and expresses that 1 has been divided

b

into b equal parts and that a such parts are taken.

The quantity a, written above the line, is called the numer ator; the quantity b, written below the line, the denominator; and both are called terms of the fraction.

1

b

One of the equal parts, as is called the fractional unit; and generally, the reciprocal of the denominator is the frac tional unit.

The numerator always expresses the number of times that the fractional unit is taken; for example, in the given fraction, the 1 fractional unit

b

is taken a times.

63. An entire quantity is one which does not contain any fractional terms; thus,

a2bcx is an entire quantity.

A mixed quantity is one which contains both entire and frac tional terms; thus,

a2b+ — is a mixed quantity.

Every entire quantity can be reduced to a fractional form having a given fractional unit, by multiplying it by the denominator of the fractional unit and then writing the product over the denominator; thus, the quantity c may be reduced to a fractional

1

form with the fractional unit, by multiplying e by 5 and

dividing the product by b, which gives

bc

64. If the numerator is exactly divisible by the denominator, a fractional expression may be reduced to an entire one, by simply performing the division indicated; if the numerator is not exactly divisible, the application of the rule for division will sometimes reduce the fractional to a mixed quantity.

65. If the numerator a of the fraction

aq

α "

b

be multiplied by

any quantity, q, the resulting fraction will express 9 times

b

a

as many fractional units as are expressed by ; hence:

b

Multiplying the numerator of a fraction by any quantity is equivalent to multiplying the fraction by the same quantity.

66. If the denominator be multiplied by any quantity, q, the value of the fractional unit, will be diminished 9 times, and the

α

resulting fraction will express a quantity q times less than qb

the given fraction; hence:

Multiplying the denominator of a fraction by any quan'ity, is equivalent to dividing the fraction by the same quantity.

67. Since we may multiply and divide an expression by the same quantity without altering its value, it follows from Arts 65 and 66, that :

Both numerator and denominator of a fraction may be multiplied by the same quantity, without changing the value of the fraction. In like manner it is evident that:

Both numerator and denominator of a fraction may be divided by the same quantity without changing the value of the fraction.

68. We shall now apply these principles in deducing rules for the transformation or reduction of fractions.

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