ON THE

ADDITION, SUBTRACTION, MULTIPLICATION,

AND

DIVISION OF ALGEBRAIC QUANTITIES.

§ I. Addition of Algebraic Quantities.

54. The addition of algebraic quantities is performed by connecting those that are unlike with their proper signs, and collecting those that are like into one sum; for the more ready effecting of which, it may not be improper to premise a few propositions, from which all the necessary rules may be derived.

55. If two or more quantities are like, and have like signs, the sum of their coefficients prefixed to the same letter, or letters, with the same sign, will express the sum of these quantities.

Thus, 5a added to 7a is 12a;

=

And -5a added to -3a is =

- 8a.

For, if the symbol a be made to represent any quantity or thing, which is the object of calculation, 5a will represent five times that thing, and 7a seven times the same thing, whatever may be the denomination or numeral value of a; and consequently, if the quantities 5a and 7a are to be incorporated, or added together, their sum will be twelve times the thing denoted by a, or 12a.

Moreover, since a negative quantity is denoted by the sign of subtraction thus, if a+b=a-c, bc, and c-b. A debt is a negative kind of property, a loss a negative gain, and a gain a negative loss.

Therefore it is plain that the quantities, - 5a and -3a, will produce, in any mixed operation, a contrary effect to that of the positive quantities with which they are connected; and consequently, after incorporating them in the same manner as the latter, the sign must be prefixed to the result; so that if a be greater than a, it is evident that 5 (▲ − a)+3 (-a), or (5A -- 5a)+(3s-3a)=8A-8a; and therefore the sum of the quantities -5a and -3a, when taken in their isolated state, will, by a necessary extension of the proposition, be-8a.

56. If two quantities are like, but have unlike signs, the difference of their coefficients, prefixed to the same letter, or letters, with the sign of that which hath the greater coefficient, will express the sum of those quantities.

Thus 6a added to -4a is

And 6a added to +4a is

+2a; =- - 2a.

Since, Art. (36), the compound quantity a-b+c-d, &c. is positive or negative, according as the sum of the positive terms is greater or less than the sum of the negative ones, the aggregate orsum of the quantities 4a-2a+2a-2a, or 6a -4a, will be +2a: since the sum of the positive terms is greater than the sum of the negative ones. And the sum of the quantities a-4a+3a-2a, or 4a - 6a, will be -2a: since the sum of the negative terms is greater than the sum of the positive

ones:

Corollary. Hence it appears, that if the sum of the positive terms be equal to the sum of the negative ones, their aggregate or sum will be nothing. Thus 5a-5a=0; and 5a ·3a4a6a-9a-9a=0.

57. The preceding proposition is demonstrated in the following manner by BONNYCASTLE in his Algebra. Vol. II. 8vo.

Where the quantities are supposed to be like, but to have unlike signs, the reason of the operation will readily appear, from considering that the addition of algebraic quantities, taken in a general sense, or without any regard to their particular values, means only the uniting of them together, by means of the arithmetical operations denoted by the signs + and; and as these are of contrary, or opposite natures, the less quantity must be taken from the greater, in order to obtain the incorporated mass, and the sign of the greater prefixed to the result. So that if 6a is to be added to 4(-a), or to 4A-4a, the result will evidently be 4+6a-4a, or 41+ 2a; and if 4a is to be added to 6(A-a), or to 6A-6a, the result will be 6A+4a-6a, or 6A-2a; whence, by making this proposition general, as in the last, the sum of the isolated quantities 6a and -4a will be +2a, and that of 4a and -6a will be -2a.

58. If two or more quantities be unlike, their sum can only be expressed by writing them after each other, with their proper signs.

Thus, the sum of 2a and 2b, can only be expressed, with the sign+between them, which denotes that the operation of addition is to be performed when we assign values to a and b.

For, if a 10, and b=5; then the sum of 2a and 26 can be neither 4a nor 46, that is, neither 4×10-40, nor 4×5=20; but 2x10+2x5=20+10=30. In like manner, the sum of 3a, 56, 2c, and -8d, can no otherwise be incorporated, or added together, than by means of the signs+ and -; thus, 3a-56-2c-8d.

These propositions being well understood, the following practical rules, for performing the addition of algebraic quantities, which is generally divided into three cases, are readily deduced from them.

CASE I.

When the quantities are like, and have like signs.

RULE.

59. Add all the numeral coefficients together, to their sum prefix the common sign when necessary, and subjoin the common quantities, or letters.

In this example, in adding up the first column, we say, 1+ 9+1+7+3+2=23, to which the common letter x is subjoined. It is not necessary to prefix the sign to the result, since the sign of the leading term of any compound algebraic expression, when it is positive, is seldom expressed; for (14) when a quantity has no sign before it, the sign + is always understood. And it may be observed when it has no numeral coefficient, unity or 1 is always understood.

Also, the sum of the second column is found thus, 8+1+9 +1+4+3=26, to which the sign is prefixed, and the. common letter a annexed.

Again, the sum of the third column is found thus; 3+1+ 9+7+1+4=25, to which the sign is prefixed, and the

common letter b subjoined. So that the sum of all the quantities is expressed by 23 times x plus 26 times a minus 25 times b..

Ans. 19x-15a.

Ex. 4. Add together 2x+3a, 4x+a, 5x+8a, 7x+2a, and x+a. Ex. 5. Add together 7x-5bc, 3x2-bc, x2-4bc, 5x3bc, and 4x2-4bc. Ans. 20x215bc. Ex. 6. Required the sum of 3x3+4x2—x, 2x31+x2—3x, 7x+2x3-2x, and 4x+2x2-3x. Ans. 16x9x2-9x. Ex. 7. What is the sum of 7a3-3a2b+2ab2 - 363, ab2 a2 b-b3+4a3, —5b5ab3--4a3b+6a3, and -a2 b+4ab?— 4b+a3 ? Ans. 18a3-9a2b+12ab2-13b3.

2

Ex. 8. Add together 2x2y-x+2, x2y-4x+3, 4x2y-3x +1, and 5x2y-7x+7. Ans. 12x2y—15x+13.

1

Ex. 9. Required the sum of 30-13--3xy, 23—10x2 —4xy, — 14x2 -7xy+14,—5xy+10-16x2, and 1-2x2

xy

1

1

1

1

Ans. 78-55x2—20xy. Ex. 10. Add 3(x+y)2-4(a—b)3, (x+y)2 —(a—b), —7 (a−b)3+5(x+y)3, and 2(x+y)2 —(a—b)3 together.

Ans. 11(x+y)2-13(a—b)3.

CASE II.

When the quantities are like, but have unlike signs.

RULE.

60. Add all the positive coefficients into one sum, and those that are negative into another; subtract the lesser of these sums from the greater; to this difference, annex the common letter or letters, prefixing the sign of the greater, and the result will be the sum required.

In adding up the first column, we say 3+9+7=+19, and (5+1+4)=-10; then, +19-10-+9= the aggregate sum of the coefficients, to which the common quantity 23 is annexed.

In the second column, the sum of the positive coefficients is 3+6+1=10, and the sum of the negative ones is −(5+2 +3)-10; then, 10-10=0; consequently, (by Cor. Art. 56), the aggregate sum of the second column is nothing. And in the third column, the sum of the positive coefficients is 6+7+3=16, and the sum of the negative one is (6+9+ 4)=-19; then +16-19=-3; to which the common letter is annexed.

Ex. 2. 5x2-6a+4x-3 -2x2+ a—9x+7 7x+7a+7x-1 -x-3a-2x+3 +3x2+ a-4x+4

-7x-4a+3x-5

522-4a-x+5

Ex. 4.

3(a+b)* — 5(x2+y2)2+3(a2+c2)3+9xy (a+b)*+ (x2+y3)3—5(a3—c2)3—4.xy +8(a+b)* — 6(x2+y3)2+8(a3+c2)3+ xy —2(a+b)* — (x2+y3)2—7(a3+c2)3 — 3xy +5(a+b)3 — 7(x2+y3)2 — . (a3+c2)3— xy

1

13(a+b)*—18(x2+y2)2—2(a3+c2)2+2xy