MULTIPLICATION. (28.) In the multiplication of algebraic quantities, four circumstances are to be considered. 1. The signs of the quantities: 2. Their co-efficients : 3. The letters of which they are composed; and 4. The indices or exponents of those letters. (29.) In performing any operation in multiplication, we must, therefore, observe the four following rules. 1. When quantities having like signs are multiplied together, the product will be +. On the contrary, if their signs are unlike, the sign of the product will be —.* 2. That the co-efficients of the factors must be multiplied together, to form the co-efficient of the product. 3. That the letters of which the factors are composed must be set down, one after another, according to their order in the alphabet. 4. That if the same letter be found in both factors, the indices of this letter must be added, to form its index in the product. That like signs make +, and unlike signs in the product, may be il lustrated thus: First. When a is to be multiplied by + b, this denotes that + a is to be taken as many times as there are units in b; and because the sum of any number of affirmative terms is affirmative, it is obvious that + ax + b ← + ab. - - ab. -- Secondly.—If two quantities are to be multiplied together, the result will be actually the same, in whatever order they are placed: for a times b is the same as b times a; and, therefore, when a is to be multiplied by + b, or +b by — a, it is the same thing as taking - -a as many times as there are units in ÷ b; and as the sum of any number of negative terms is negative, it is plain that a × + b, or + b × — a = ~ Lastly. When a is to be multiplied by — b, we have ab for the product at first sight, but still we must determine whether the sign + or – is to be placed before the product. Now it cannot be the sign for + X, or which is the same thing, + a xb gives ab, and - a by 2 cannot produce the same result as a x+b; but must produce a contrary result, to wit, +ab; consequently we have the following rule: multiplied by -produces +, in the same manner as + × + give +, But this illustration may be demonstrated thus: When the compound quantity + ab is to be multiplied by + c, we repeat or add ab to itself as often as there are units in c; hence, since the sum of any number of affirmative terms is affirmative, and the sum of any number of negative terms is negative, it is obvious, that ab multiplied by+c produces + ac- - be; for the same reason, + a — b multiplied by+d produces + ad — bt. Whence, if from times (a - b) + ad bd + ac be - -= + uc + bd be ad like signs produce plus unlike signs produce minus. }....+2x-3..... and letters.}-3ab × + 5 cd = signs, co-efficients, 15 abed. letters, and indices. } -4 a2 l2 × — 3a bd3 = + 12 a° ba d3. Note. From the division of algebraic quantities into simple and compound, three cases of multiplication arise; and in performing the operation in all these cases, we must attend first to the signs, then the co-efficients, and lastly the letters and indices. CASE I. When both the factors are simple quantities. Rule. Attend to the signs, co-efficients, and indices, by the foregoing Rules (29 and 30). CASE II. When one factor is compound, and the other simple. Rule. Multiply each term of the compound factor by the simple factor, as in the last case, and the result will be the product required. CASE III. When both factors are compound quantities. Rule. Multiply each term of the multiplicand by each term of the multiplier; and, then, placing like quantities under each other, the sum of all the terms will be the product required. Scholium. We may substitute for a and b (Ex. 1.) any determinate numbers; so that the above example will furnish the following (31.) THEOREM. The square of the sum of any two numbers is equal to the sum of their squares, together with twice their product. From Example 2 we have another (32) THEOREM. The product of the sum of two numbers multiplied by their difference, is equal to the difference of the squares of those numbers. And from this may be derived a third (33.) THEOREM. The difference of two square numbers is always a product, and divisible both by the sum, and by the difference of the roots of those two squares, consequently, the difference of two squares can never be a prime number. Note-All numbers, such as 2, 3, 5, 7, &c. which cannot be represented by factors, are called simple or prime numbers; whereas, others, as 4, 6, 8, 9, &c. which may be represented by the factors 2 × 2, 2 × 3, 2 × 4, or 2 X 2 X 2, and 3 × 3, are called composite numbers. Miscellaneous Examples. 1. Multiply 10ac by 2a. 3. Multiply 3a-26 by 3b. 8. Multiply 3a +26 by 3a - 2b. 4. Multiply xxy+ys by x + y. 8. Multiply 3a2 2ax +5x1 by 3a2 ·4ax-7 x2. 9. Multiply 3x + 2x2y2 + 3y3 by 2ư3 — 3x2y2 + 3y. 10. Multiply a + ab + b2 by a -26. DIVISION. (34.) The same circumstances are to be taken into consideration, in the division of algebraic quantities, as in their multiplication, and consequently the four following rules must be attended to. 1. That if the dividend and divisor be like, then the sign of the quotient will be + if on the contrary unlike, then the sign of the quotient will be 2. That the co-efficient of the dividend is to be divided by the co-efficient of the divisor, to obtain the co-efficient of the quotient. 3. That all the letters common to both the dividend and divisor, must be rejected in the quotient. 4. That if the same letter be found in both the dividend and divisor, with different indices, then the index of that letter in the divisor, must be subtracted from its index in the dividend, to obtain its index in the quotient. (35.) From these rules we derive the four following expressions :1st.abc divided by + ac, or + abc = + b + ac 2d. +6 abc.... by 3bc -2 a Note. Of Division, also, there are three cases, as in Multiplication. CASE I. When the dividend and divisor are both simple terms. Rule. Divide the co-efficient of the numerator by the co-efficient of the denominator, expunge those letters which are common to both terms of the fraction, and add the odd ones, if any, to the numeral quotient (Art. 34 and 35.) CASE II. When the dividend is a compound quantity, and the divisor a simple one. Rule. Divide each term of the dividend separately by the imple divisor, and the resulting quantities will be the quotient required (35) 5th. Divide 14ab7a9 21a2b2+ 42 a3b by 7ab. CASE III. When the dividend and divisor are both compouna quantities. Rule I. Arrange both dividend and divisor according to the powers of the same letter, beginning with the highest. 2. Then find how often the first term of the divisor is contained m the first term of the dividend, and place the result in the quotient. 3. Multiply each term of the divisor by this quantity, and place the product under the corresponding (i. e. like) terms in the dividend, and then subtract the one from the other. 4. To the remainder bring down as many terms of the dividend as will make its number of terms equal to the number of terms in the divisor; and then proceed as before, till the terms are all brought down as in common arithmetic. Illus. In this example, the dividend is arranged according to the powers of a, the first term of the divisor. Having done so, we proceed by the following steps, which sufficiently illustrate the rule. 1. a is contained in a3, a2 times; and this (a2) is put in the quotient. 2. ab is multiplied by a2, and it gives a3 3. From as or a3, ua b. 3a2b in the dividend, we subtract the product of (a — b) a2 a2b, and the remainder is -2a2b. 4. The next term + 3 ab2 in the dividend is now brought down. 5. a is contained in — 2 aa b, 2ab times, which we put in the quotient. 6. Multiply and subtract as before, and the remainder is ab2. 7. Bring down the last term of the dividend — b3. 8. a is contained in ab2, + ba times; put this in the quotient. 9. Multiply and subtract as before, and nothing remains. The quotient, therefore, is a2 - 2 ab + b2, and we call it the ansu er. 2ab+b2 by the 10. To prove the operation, multiply the quotient aa divisor ab, and the product will be the dividend; for by the nature of Division it is evident, that if one number be divisible by another, the quotient multiplied by the divisor will produce the dividend. Whence the reason of the above rules (Art. 34) and operations is evident from what was proved in Multiplication. (See Art. 28 and 29.) |