In general, to reduce a monomial fraction, we have the following

RULE.

Suppress all the factors common to the numerator and denominator, and write those letters which are not common, with their respective exponents, in the term of the fraction which contains them.

In the last example, as all the factors of the dividend are found in the divisor, the numerator is reduced to unity; for, in fact, both terms of the fraction are divided by the numerator.

52. It often happens, that the exponents of certain letters, are the same in the dividend and divisor.

is a case in which the letter b is affected with the same exponent in the dividend and divisor: hence, it will divide out, and will not appear in the quotient.

But if it is desirable to preserve the trace of this letter in the quotient, we may apply to it the rule for the exponents (Art. 50) which gives

This new symbol bo, indicates that the letter b enters 0 times as a factor in the quotient (Art. 13); or what is the same thing, that it does not enter it at all.. Still, the notation shows that b was in the dividend and divisor with the same exponent, and has disappeared by division.

15a2b3c2

In like manner,

3a2bc2

5a0b2c05b2.

53. We will now show that the power of any quantity whose exponent is 0, is equal to unity. Let the quantity be represented by a, and let in denote any exponent whatever.

=am―m= : ao, by the rule of division.

Then,

am

am

1, since the numerator and denominator are equal:

: 1, since each is equal to

We observe again, that the symbol ao is only employed conventionally, to preserve in the calculation the trace of a letter which entered in the enunciation of a question, but which may disappear by division.

Division of Polynomials.

54. The object of division, is to find a third polynomial called the quotient, which, multiplied by the divisor, shall produce the dividend.

Hence, the dividend is the assemblage, after reduction, of the partial products of each term of the divisor by each term of the quotient, and consequently, the signs of the terms in the quotient must be such as to give proper signs to the partial products.

Since, in multiplication, the product of two terms having the same sign is affected with the sign +, and the product of two terms having contrary signs, with the sign, we may conclude,

1st. That when the term of the dividend has the sign+, and that of the divisor the sign of +, the term of the quotient must have the sign +.

2d. When the term of the dividend has the sign +, and that of the divisor the sign the term of the quotient must have -; because it is only the sign, which, combined with

the sign

can produce the sign of the dividend.

the sign 3d. When the term of the dividend has the sign and that of the divisor the sign +, the quotient must have the sign

That is, when the two corresponding terms of the dividend and divisor have the same sign, their quotient will be affected with the sign +, and when they are affected with contrary signs, their quotient will be affected with the sign ; again, for the sake of brevity, we say that

We first divide the term a2 of the dividend by the term a of the divisor, the partial quotient is a, which we place under the divisor. We then multiply the divisor by a, and subtract the product a2 ax from the dividend, and to the remainder bring down x2. We then divide the first term of the remainder, by a, the quotient is XC. We then multiply the divisor by -x, and, subtracting as before, we find nothing remains. Hence, ax is the exact quotient.

- ax,

SECOND EXAMPLE.

Let it be required to divide 26a2b2 + 10a1

48a3b24ab3 by 4ab 5a2 + 362. In order that we may follow the steps of the operation more easily, we will arrange the quantities with reference to the letter a.

It follows from the definition of division, and the rule for the multiplication of polynomials (Art. 44), that the dividend is the assemblage, after addition and reduction, of the partial products of each term of the divisor, by each term of the quotient sought. Hence, if we could discover a term in the dividend which was derived, without reduction, from the multiplication of a term of the divisor by a term of the quotient, then, by dividing this term of the dividend by that of the divisor, we should obtain a term of the required quotient.

Now, from the third remark of Art. 45, the term 10a, affected with the highest exponent of the letter a, is derived, without reduction from the two terms of the divisor and quotient, affected

with the highest exponent of the same letter. the term 10a1 by the term 5a2, we shall required quotient.

Dividend.

48a3b26a2b2 + 24ab3

10a4 +10a4

Hence, by dividing have a term of the

Divisor.

Since the terms 10a and 5a2 are affected with contrary signs, their quotient will have the sign; hence, 10a, divided by - 5a2, gives 2a2 for a term of the required quotient. After having written this term under the divisor, multiply each term of the divisor by it, and subtract the product,

from the dividend, which is done by writing it below the dividend, conceiving the signs to be changed, and performing the reduction. Thus, the remainder after the first partial division is

- 40a3b + 32a2b2 + 24ab3.

This result is composed of the partial products of each term. of the divisor, by all the terms of the quotient which remain to be determined. We may then consider it as a new dividend, and reason upon it as upon the proposed dividend. We will therefore divide the term - 40a3b, affected with the highest exponent of a, by the term - 5a2 of the divisor. Now, from the preceding principles,

for a new term of the quotient, which is written on the right of the first. Multiplying each term of the divisor by this term of the quotient, and writing the products underneath the second dividend, and making the subtraction, we find that nothing remains. Hence,

is the required quotient, and if the divisor be multiplied by it, the product will be the given dividend.

By considering the preceding reasoning, we see that, in each partial operation, we divide that term of the dividend which is

affected with the highest exponent of one of the letters, by that term of the divisor affected with the highest exponent of the same letter. Now, we avoid the trouble of looking out these terms by writing, in the first place, the terms of the dividend and divisor in such a manner that the exponents of the same letter shall go on diminishing from left to right.

This is what is called arranging the dividend and divisor with reference to a certain letter. By this preparation, the first term on the left of the dividend, and the first on the left of the divisor, are always the two which must be divided by each other in order to obtain a term of the quotient.

55. Hence, for the division of polynomials we have the following

RULE.

I. Arrange the dividend and divisor with reference to a certain letter, and then divide the first term on the left of the dividend by the first term on the left of the divisor, for the first term of the quotient; muliply the divisor by this term and subtract the prod uct from the dividend.

II. Then divide the first term of the remainder by the first term of the divisor, for the second term of the quotient; multiply the divisor by this second term, and subtract the product from the result of the first operation. Continue the same process, and if the remainder is 0, the division is said to be exact.

Divide

5y2 8x2

THIRD EXAMPLE.

21x3y2+25x2y3 + 68xy1 — 40y5-56x518x1y by 6xy.

− 40y3 + 68xy1 +25x2y3 + 21x3y2 — 18x1y — 56x5 ||5y2 — 6xy — 8x2