A letter, which is multiplied once by itself, or which has two for an exponent, is said to be raised to the second power·· A letter which is multiplied twice successively by itself, or which has 3 for an exponent, is said to be raised to the third power. In general, the power of a letter is designated according to the figure, which it has for an exposent, thus a with 7 for an exponent is called the seventh power of a.

A letter which has no exponent is considered as having unity for its exponent, thus, a is the same as a1.

From what has been said, it will be perceived, that in order to raise a letter to a given power, it is necessary to multiply it successively by itself as many times less one, as there are units in the exponent of this power.

5. Let it next be required to multiply a3 by a3. According to no. 1 the product will be expressed by a3 a. In this product the letter a, it will be observed, occurs three times as a factor and also five times as a factor, whence on the whole it is found eight times as a factor. The product a3 a3 may therefore according to no. 4 be expressed more concisely, thus, a3. In like manner the product of a' by a3 will be a1. Whence, in general, The product of two powers of the same letter will have for an exponent the sum of the exponents of the multiplier and multiplicand.

6. Let it be proposed next to multiply a b c by a* b3 c2 d. According to no. I the product will be a b c a b c d, or by no. 2. a3 a1 b2 b3 c c2 d; but this expression may be reduced by the rule just given to a b c d; whence

as b2 cx a b3 c2 da b3 c3 d.

From what has been done we have the following rule for the multiplication of simple quantities, viz. 1o. Multiply the coefficients together; 2°. write in order in the product thus obtained the letters which are found at once in both the multiplier and multiplicand, observing to give to each letter the sum of the exponents, with which this letter is affected in the two factors; 3o. if a letter is found in one of the factors only, write it in the product with the exponent which it has in this factor.

36. Let us pass to the multiplication of polynomials.

To indicate that a polynomial a+b, for example, is multiplied by another c+d, we draw a vinculum over each and connect them by the sign of multiplication, thus,

a + bx c + d,

or, which is the better method, we inclose each of the quantities in a parenthesis and write them in order one after the other, either with or without the sign of multiplication, thus,

(a + b) x (c + d), or (a+b) (c + d).

1. To multiply a + b by c.

To form the product required, it is evident, that we must take c times each of the parts a and b of which the quantity a + b is composed.

The product of multiplied by

a + b

C

is therefore

In like manner

multiplied by gives

a c + b c

2a + b2 c + d

h

Qah + b2 ch + dh

Since a

bis smaller than a

2. To multiply ab by c. by the quantity b, a e the product of a by c, it is evident, will be too large for the product required by b times c or bc; whence to obtain the true result, from a c we must subtract b c. The product of a b multiplied by

is therefore

In like manner multiplied by gives

с

a c bc

a2 + c2 — d h — ef

ah

a3h+ah c2 — a dh'-ahef From what has been done it is evident that, If two terms each affected with the sign + be multiplied together, the product must have the sign + ; but if one of the terms be affected with the sign + and the other with the sign —, the product must have the

3. Let it be proposed next to multiply ab by cd. In this case, it is evident, that, if we take c times a

sult will be too great by d times a true product, from c times a

b the re

-b; whence to obtain the

b or a c

bc we must subtract

ac-be-ad+b d

From this example it appears that, If two terms be affected each with the sign —, the product of these terms should be affected with the sign +.

If in the expression of a product there occur similar terms, the expression may be abridged by uniting these terms into

=

To verify this result let a 5, b=2, c=3.

From what has been done we have the following rule for the multiplication of polynomials, viz. 1°. Multiply each term of the multiplicand by each term of the multiplier, observing with respect to the signs, that if two terms multiplied together have each the same sign, the product must have the sign +, but if they have different signs, the product must have the sign; 2°. add together the partial products thus obtained, taking care to unite in one, terms which are similar.

37. A polynomial is said to be arranged with reference to some letter, when its terms are written in order according to the powers of this letter. The polynomial

a2b3 + a3 b — a ba + a1 b2

for example, arranged in descending powers of the letter a stands thus, a* b2 + a3 b + a2 b3 — a ba, arranged in ascending powers of the letter b it stands thus, a3 b + a b2 + a2 b3 — a ba.

The letter with reference to which the arrangement is made is called the principal letter.

To facilitate the multiplication of polynomials, it is usual, 1o. to arrange the quantities to be multiplied according to the powers of the same letter; 2°. to dispose of the partial products in such a manner that those terms, which are similar, shall fall under each other. Let it be proposed, for example, to multiply

b3 + b2 a + a2 + ba3 by 4 b2-3ba+3a2.

The multiplier and multiplicand being both arranged with reference to the letter a, the work will be as follows

3 a + 4 b2 a3 + 4 b3 a2 + b2 a + 4 b3

38. The following examples will serve as an exercise in the multiplication of polynomials.

66

Answer 20 a 41 a* b + 50 a3 b2 — 45 a2 b3 + 25 a b1 — 6 b

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39. A term, which contains one literal factor only, is said to be of the first degree; a term, which contains two literal factors only, is said to be of the second degree, &c. In general, the degree of a term is marked by the number, which expresses the sum of the exponents of the letters, which enter into this term. The coefficient is not reckoned in estimating the degree of the term. Thus a b c is a term of the 6th degree, and 7 a b3 is a term of the fourth degree,

A polynomial is said to be homogeneous, when all its terms are of the same degree. Thus, 3 a2 — 4 a b, 5 a3 + a b c — b' are homogeneous polynomials.

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40. From the rules for multiplication, which have been laid down, it follows,

1o. If the polynomials proposed for multiplication are each homogeneous, the product of these polynomials will also be homogeneous, and the degree of each term of the product will be equal to the sum of the degrees of any two terms whatever of the multiplier and multiplicand. Thus, in the first example, art. 38, all the terms of the multiplicand being of the third degree and those of the multiplier of the second degree, all the terms of the product are of the fifth degree. When therefore the factors of a product are homogeneous, we may readily detect by means of this remark any error in regard to the exponents, which may have occurred in the course of the work.

2o. In the multiplication of polynomials, if there be no reduction of similar terms, the number of terms in the product will be equal to the number of terms in the multiplicand multiplied by the number of terms in the multiplier. Thus, if there be 5 terms in the multiplicand and 4 in the multiplier, there will be 20 in the product.

3o. But if there be a reduction of similar terms, then the number of terms in the product may be much less. It should be observed, however, that among the different terms of the product there will be two at least, which will not admit of reduction with any other, viz. 1°. The term arising from the multiplication of the term in the multiplicand affected with the highest exponent of one of the letters, by the term in the multiplier affected with the highest exponent of the same letter. 2o. The term arising from the multiplication of the two terms affected with the lowest exponent of the same letter.

The manner in which an algebraic product is formed by means of its factors is called the law of this product. This law, it will readily be perceived, remains always the same, whatever may be the values attributed to the letters which enter into the factors.