33. From the manner of proceeding in multiplication, it is evident, that if all the terms of the multiplicand are of the same degree (27), and those of the multiplier are also of the same degree, all the terms of the product will be of a degree denoted by the sum of the numbers, which mark the degree of the terms of each of the factors.

In the first example, the multiplicand is of the fourth degree, the multiplier of the third; and the product is of the seventh. In the second example, the multiplicand is of the sixth degree, the multiplier of the third; and the product is of the ninth.

Expressions of the kind just referred to, the terms of which are all of the same degree, are called homogeneous expressions. The above remark, with respect to their products, may serve to prevent occasional errors, which one may commit by forgetting some of the factors in the several parts of the multiplication.

54. Algebraic operations performed upon literal quantities, as they permit us to see how the several parts of the quantities concur to form the results, often make known some general properties of numbers independent of every system of notation. The multiplications that follow, lead to conclusions of the greatest importance, and of frequent use in the subsequent parts of this work.

It appears from the first of these products, that the quantity a+b, multiplied by a-b, gives a2-b2; whence it is evident that, if we multiply the sum of two numbers by their difference, the product will be the difference of the squares of these numbers.

If we take, for example, the sum 11 of the numbers 7 and 4,

and multiply it by the difference 3 of these numbers, the product 3 x 11, or 33, will be equal to the difference between 49, the square of 7, and 16, the square of 4.

By the second example, in which a+b is twice a factor, we learn; that the second power, or the square of a quantity composed of two parts a and b contains the square of the first part, plus double the product of the first part by the second, plus the square of the second.

The third example, in which we have multiplied the second power of a + b by the first, shows; that, the third power or cube of a quantity composed of two parts contains the cube of the first, plus three times the square of the first multiplied by the second, plus three times the first multiplied by the square of the second, plus the eube of the second.

35. As we have often occasion to decompose a quantity into its factors, and as the algebraic operations are dispensed with, when it can be done, in order to exhibit the formation of the quantities to be considered, as distinctly as possible, it is necessary to fix upon some signs proper to indicate multiplication between complex quantities.

We use indeed the marks of a parenthesis to comprehend the factors of a product. The expression

5 a

(5a1 — 3 a2 b2 + b1) (4 a b2 — a c2 + d3) (b2 — c2), for example, indicates the product of the compound quantities 3 a2 b2 +b4, 4 ab2 — ac2 + d3 and b2 — C2. Bars were used formerly by some authors placed over the factors thus,

5 a4 ·3 a2 b2 + b4 × 4 a b2 — a c2 + d3 × b2 — c2; but as these may happen to be too long or too short, they are liable to more uncertainty than the marks of a parenthesis, which can never admit of any doubt with respect to the quantity belonging to each factor. They have accordingly been preferred.

of the division of algebraic quantities.

56. ALGEBRAIC division, like division in arithmetic, is to be regarded as an operation disigned to discover one of the factors of a given product, when the other is known. According to this definition, the quotient multiplied by the divisor must produce anew the dividend.

By applying what is here said to simple quantities we shall see by art. 21, that the dividend is formed from the factors of the divisor and those of the quotient; whence, by suppressing in the dividend all the factors which compose the divisor, the result will be the quotient sought.

Let there be, for example, the simple quantity 72 as b3 c2 d to be divided by the simple quantity 9a3bc2; according to the rule above given, we must suppress in the first of these quantities the factors of the second, which are respectively

9, as, b, and c2.

It is necessary then, in order that the division may be performed, that these factors should be in the dividend. Taking them in order, we see in the first place that the coefficient 9 of the divisor, ought to be a factor of the coefficient 72 of the dividend, or that 9 ought to divide 72 without a remainder. This is in fact the case, since 729 x 8. By suppressing then the factor 9, there will remain the factor 8 for the coefficient of the quotient.

It follows moreover, from the rules of multiplication (25), that the exponent 5 of the letter a in the dividend, is the sum of the exponents belonging to the divisors and quotient; this last exponent therefore will be the difference between the two others, or 5 32. Thus the letter a has in the quotient the exponent 2. For the same reason, the letter b has in the quotient an exponent equal to 3-1, or 2. The factor c2 being common to the dividend and divisor is to be suppressed, and we have

for the quotient required.

8 a2 b2 d

The same will apply to every other case; we conclude then, that, in order to effect the division of simple quantities, the course to be pursued is,

To divide the coefficient of the dividend by that of the divisor ;

To suppress in the dividend the letters which are common to it and the divisor, when they have the same exponent; and when the exponent is not the same, to subtract the exponent of the divisor from that of the dividend, the remainder being the exponent to be affixed to the letter in the quotient;

To write in the quotient the letters of the dividend which are not in the divisor.

37. If we apply the rule now given for obtaining the exponent of the letters of the quotient, to a letter which has the same

4. That a term having the sign, --, multiplied by a term having the sign, gives a product which has the sign +.

It is evident from this table, that when the multiplicand and multiplier have the same sign, the product has the sign +, and that when they have different signs, the product has the sign —.

To facilitate the practice of the multiplication of polynomials, I have subjoined a recapitulation of the rules to be observed. 1. To determine the sign of each particular product according to the rule just given; this is the rule for the signs.

2. To form the coefficients by taking the product of those of each multiplicand and multiplier (22); this is the rule for the coefli

cients.

3. To write in order, one after the other, the different letters contained in each multiplicand and multiplier (21); this is the rule for the letters.

4. To give to the letters, common to the multiplicand and multiplier, an exponent equal to the sum of the exponents of these letters in the multiplicand and multiplier (25); this is the rule for the exponents.

32. The example below will illustrate all these rules. Multiplicand 5a42a3b+ 4 a 2 b2 Multiplier

Several products.

Result reduced

a3

5 a7

· 4 a2 b + 2 b3

2 a b + 4 a5 b2

5a7-22ab+12a5b2-6a4b34a3b*+8a2b5.

The first line of the several products contains those of all the terms of the multiplicand by the first term a3 of the multiplier'; this term being considered as having the sign +, the products which it gives have the same signs as the corresponding terms of the multiplicand (31).

The first term 5 a* of the multiplicand having the sign plus, we do not write that of the first term of the product, which would be+; the coefficient 5 of a being multiplied by the coefficient 1 of a3, gives 5 for the coefficient of this product; the sum of the two exponents of the letter a is 4+ 3, or 7, the first term of the product then is 5 a7.

a3

The second term - 2 a b of the multiplicand having the sign the product has the sign minus; the coefficient 2 of a3 b mul

tiplied by the coefficient 1 of a3, gives 2 for the coefficient of the product; the exponent of the letter a, common to the two terms which we multiply, is 3 + 3, or 6, and we write after it the letter b, which is found only in the multiplicand. The second term of the product then is 2 a b.

The third term +4a2 b2 gives a product affected with the sign +, and by the rules applied to the two preceding terms, we find it to be + 4 a

b2.

--

The second line contains the products of all the terms of the multiplicand by the second term 4 a2 b of the multiplier. This last having the sign, all the products which it gives must have the signs contrary to those of the corresponding terms of the multiplicand; the coefficients, the letters, and the exponents are determined as in the preceding line.

The third line contains the products of all the terms of the multiplicand by the third term +263 of the multiplier. This term having the sign+, all the products which it gives have the same sign as the corresponding terms of the multiplicand.

After having formed all the several products which compose the whole product, we examine carefully this last, to see whether it does not contain similar terms; if it does, we reduce them according to the rule (19), observing that two terms are similar, which consist of the same letters under the same exponents. In this example there are three reductions, viz;

These reductions being made, we have for the result the last line

of the example.

See another example to exercise the learner, which is easily performed after what has been said.