51. Let us first consider the case in which both dividend and divisor are monomials. Take

35a5b2c2 to be divided by Ta2bc;

The operation may be indicated thus,

Now, since the quotient must be such a quantity as multiplied by the divisor will produce the dividend, the co-efficient of the quotient multiplied by 7 must give 35; hence, it is 5.

Again, the exponent of each letter in the quotient must be such that when added to the exponent of the same letter in the divisor, the sum will be the exponent of that letter in the dividend. Hence, the exponent of a in the quotient is 3, the exponent of bis 1, that of c is 1, and the required quotient is 5a3bc.

Since we may reason in a similar manner upon any two monomials, we have for the division of monomials the following

RULE.

1. Divide the co-efficient of the dividend by the co-efficient of the divisor, for a new co-efficient.

11. Write after this co-efficient, all the letters of the dividend, and give to each an exponent equal to the excess of its expe nent in the dividend over that in the divisor.

52. It follows from the preceding rule that the exact division of monomials will be impossible :

1st. When the co-efficient of the dividend is not divisible by that of the divisor.

2d. When the exponent of the same letter is greater in the divisor than in the dividend.

This last exception includes, as we shall presently see, the case in which the divisor has a letter which is not contained in the dividend.

When either of these cases occurs, the quotient remains under the form of a monomial fraction; that is, a monomial expression, necessarily containing the algebraic sign of division. Such expressions may frequently be reduced.

Here, an entire monomial cannot be obtained for a quotient; for, 12 is not divisible by 8, and moreover, the exponent of c is less in the dividend than in the divisor. But the expression can be reduced, by dividing the numerator and denominator by the factors 4, a2, b, and c, which are common to both terms of the fraction.

In general, to reduce a monomial fraction to its lowest terms: Suppress all the factors common to both numerator and denomi

In the last example, as all the factors of the dividend are found in the divisor, the numerator is reduced to 1; for, in fact, both terms of the fraction are divisible by the numerator.

53. 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 exponents (Art. 51), which gives

The symbol 6o, indicates that the letter b enters 0 times as a factor in the quotient (Art. 16); 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.

In like manner,

15a2b3c2
3a2bc2

=5ab2c0 = 562.

54. We will now show that the power of any quantity whose exponent is 0, is equal to 1. Let the quantity be represented by a, and let m denote any exponent whatever.

1, since the numerator and denominator are equal:

hence, a = 1, since each is equal to

am

am

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.

55. In the second place, if the dividend is a polynomial and the divisor is a monomial, we divide each term of the dividend by the divisor, und connect the quotients by their respective signs.

EXAMPLES.

Divide 6a2x4y6 — 12a3x3y6 + 15a4x5y3 by 3a2x2y2.

Ans. 2x2y4axy1 + 5a2x3y.

Divide 12a4y-16a5y5 +20a6y4-28a7y3 by -4a4y3.

Ans. 3y3+ 4ay2 — 5a2y + 7a3.

56. In the third place, when both dividend and divisor are polynomials. As an example, let it be required to divide 48a3b+24ab3 by 4ab 5a2 + 362.

26a2b2+10a4

In order that we may follow the steps of the operation more easily, we will arrange the quantities with reference to the letter a.

10a4

48a3b + 26a2b2 + 24ab3 ||— 5a2 + 4ab + 3b2

It follows from the definition of division and the rule for the multiplication of polynomials (Art. 45), that the dividend is the sum of the products arising from multiplying 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 divi sor by a term of the quotient, then, by dividing this term of the dividend by that term of the divisor, we should obtain one term of the required quotient.

Now, from the third remark of Art. 46, the term 10a*, con taining the highest power of the letter a, is derived, without reduction from the two terms of the divisor and quotient, containing the highest power of the same letter. Hence, by dividing have one term of

the term 10a by the term the required quotient.

Dividend.

- 5a2, we shall

Divisor.

5a2+4ab+ 362

- 2a2+8ab
Quotient.

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. The remainder after the first operation is — 40a3b + 32a2b2 + 24ab3.

This result is composed of the 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, which contains the highest power of a, by the term 5a2 of the divisor.

This gives

+ 8ab

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 operation, we divide that term of the dividend which contains the highest power of one of the letters, by that term of the divisor containing the highest power of the same letter. Now, we avoid the trouble of looking out these terms by arranging both polynomials with reference to a certain letter (Art. 45), which is then called the leading letter.

Since a similar course of reasoning may be had upon any two polynomials, we have for the division of polynomials 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; multiply the divisor by this term and subtract the product from the dividend.