-3d, 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.

EXAMPLE 1.

2x+3a-4b

3x+4a-b

7x+a-76

x+9a-96

9x+a- b

x+3a-3b

23x+26a-25b

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.

Ex. 4. Add together 2x+3a, 4x+a, 5x+8a, 7x +2a, and x+a.

Ans. 19x+15a. Ex. 5. Add together 7x2-5bc, 3x2-bc, x2-4bc, 5x2-bc, and 4x2-4bc. Ans. 20x2-15bc, Ex. 6. Required the sum of 3x3+4x2-x, 2x3 + x2-3x, 7x3+2x2-2x, and 4x3-2x2-3x.

Ans. 16x3+9x2-9x: Ex. 7. What is the sum of 7a3-3a2b+2ab23b3, ab2-a3b-b3+4a3, -5b3+5ab2-4a2b+6a3, and-a2b4ab3-4b3+a3?

3

Ans. 18a3-9a2b+12ab2-1363. Ex. 3. Add together 2x2y-x+2, x2y-4x+3, 4x2y-3x+1, and 5x2y-7x+7.

Ans. 12xy-15x+13.

Ex. 9. Required the sum of 30-13x2-8xy, 2310x1—4xy, —14x3—7xy+14, −5xy+10—16x3,

and 1-2x2-xy.

Ans. 78-55—20xy.

Ex. 10. Add 3(x+y)2—4(a—b)3, (x+y)2 —(a— b)3, —7(a—b)3+5(x+y)3, and 2(x+y)2—(a—b)3 together. Ans. 11(x+y)-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 x3 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.

3(a+b)3 — 5(x2+y2)2+ (a3+c2)3+9xy - (a+b)*+ (x2+y2)2—5(a3+c2)3—4xy +8(a+b)13 — 6(x2+y3)2+8(a3+c2)3+ xy —2(a+b)3 — (x2+y2)2—7(a3+c2)3—3xy +5(a+b)3— 7(x2+y3)2 — (a3+c2)3— xy

13(a+b)3 —18(x2 + y2)2—2(a3+c2)3 +2xy

Ex. 5. Required the sum of 4a3, —5a2, a2. -6a2, 9a2, and --ɑ3.

Ans. 2a.

Ex. 6. Required the sum of 4x2-3x+4, x2x2-5, 1+3x2-5x, 2x-4+7x3, 13-x-4x. Ans. 11x2-9x+9.

Ex. 7. Required the sum of 4x3-2x+y, 4x-y -x3, 9y+7x3-x, 21x-2y+9x3.

Ans. 19x3+22x+7y. Ex. 8. Required the sum of 5a3-2ab+b2, ab— 2b2-a3, b2-3ab+4a3, 4ab+2a3 —4b3.

Ans. 10a34b2.

Ex. 9. What is the sum of 2a-3x2, 5x2 -7a, -3a+x2, and a-3x2? Ans. -7a. Ex. 10. What is the sum of 4-3x, x-5, 2x-4, -4x+13, and -5x+1? Ans. 9--9x.

CASE IN:

When the quantities are unlike, or when like and unlike are mixed together.

RULE.

61. When the quantities are unlike, write them down, one after another, with their signs and coefficients prefixed; but when some are like, and others unlike, collect all the like quantities together, by taking their sums or differences, as in the foregoing cases, and set down those that are unlike as before.

EXAMPLE 1. Add together the quantities 7a3. -56, +4d, -9a, and 8c2.

Here, the quantities are all unlike; .'. (Art. 58), their sum must be written thus;

7a2-5b+4d-9a+8c2.

When several quantities are to be added together, in whatever order they are placed, their values remain the same. Thus, 7a2-5b+4d-9a + 8c2,