To add polynomials :

I. Write the quantities to be added, placing the similar terms together in separate columns.

II. Add each column, and connect the several results by their respective signs.

Ans. 9a'x.

10. Add 12ax, 5a'x, -4a'x, 6u, and -10a'x.

11. Add 4abd, -2abd', 7abd, abd', -5abd', -13abd', and 7abd. Ans. -abd'.

12. Add 2xy-2a2, 3a2+2xy, a2+xy, 4a2-3xy, and 2xy-2a2. Ans. 4a2+4xy.

13. Add 8x2x2-3xy, 5ax-5xy, 9.ry-5ax, 2a2x2+xy, and 5ax-3xy. Ans. 10a3x3-xy+5ax.

14. Add a'-2ac+cd+b, 3a2—3ac-3cd-2b, 2a'+ac-5cd+ 6b, and a-4ac+2cd-3b. Ans. 7a-8ac-5cd+2b.

15. Add 2a2x2-3mx+4m2d, 3m3d+5a2x2-5mx, 6mx-4m❜d -3a'x', and 2mx-3a2x2-3m'd.

Ans. a2x2.

16. Add 2bx-12, 3x-2bx, 5x-31/x, 31/x+12, and x2+3. Ans. 9x+3.

17. Add 106-3bx3, 2b3×3—b3, 10—2bx3, b2x2-20, and 3bx2+ Ans. 1062-2bx2+3b2x2—10.

18. Add 9bc-18ac', 15bc'+ac, 9ac-24bc', and 9ac-2.

Ans. ac- -2.

19. Add 6m2+2am+1, 6am-2m3 +4, 2m3—8am+7, and 3m2 -1. Ans. 9m2+11.

20. Add 5x-3x3+4x3-2x+10, 7x*+2x3+2x2+5x+2, and x3-3x. Ans. 12x+6x+12. 21. Add 3x'y'—5x'y'-x3y-xy+5xy, 7x'y'—4x3y+2x'y'+ 2xy+xy, and x'y'—xy'- 2x3y'+5x3y+2xy.

Ans. 6xy+8xy.

2

2

22. Add 5a+31 m2—1+4, 7a−√ m2—1—5, 3a—5√/m2—1 -8, and 2a+21 m2—1+2. Ans. 17a-Vm2—1—7. 23. What is the sum of 3a2c—2c'a+ac1‚ 2a2c3+3c2aa— 5a31⁄23, and a2c1—5c2a2+8ac? Ans. 6a2c3—4c2a3+4a3c2. 24. What is the sum of 9a(a-b)—4m√ m- -C, 7mV m-c6a(a-b), and 12 mV m-c-8a(a—b)?

Ans. 15mVm-c-5a(ab). 25. What is the sum of a+b+c+d+m, a+b+c+d—m, a+ b+c-d―m, a+b-c-d-m, and a-b-c-d-m?

Ans. 5a+3b+c-d—3m.

50. The Unit of Addition is the letter or quantity whose coefficients are added, in the operation of finding the sum of two or more quantities. Thus, in the example,

3x+2x+4x=9x

the letter x is the unit of addition. Also, in the example,

5√a+c+4√ a+c—3V a+c=6Va+c

the quantity, Va+c, is the unit of addition

51. When dissimilar terms have a common literal part, this may be taken as the unit of addition. The sum of the terms will then be expressed by inclosing the sum of the coefficients in a parenthesis, and prefixing it to the common unit.

Ans. (4a+4)y+(3c+6)x.

Add 3x+2xy, bx+cxy, and (a+b)x+2cdxy.

Ans. (a+2b+3)x+(2cd+c+2)xy.

9. Add cx+7y, 7ax-3y, and -2x+4y.

Ans. (8a-2)x+8y. 10. Add (ba)/x, and (c+2a—b)√x. Ans. (c+a)v/x. 11. Add (a+2b) m—c√m, (2a—6c) m-3a√m, (5c—4a) m— b√m, and (2a-3b)m+4a√m. Ans. (a-b-c) (m+√m). 12. Add ax+y+z, x+ay+z, and x+y+az.

Ans. (a+2) (x+y+z).

SUBTRACTION.

52. Subtraction, in Algebra, is the process of finding the difference between two quantities.

53. It is evident that 5 units of any kind or quality subtracted from 8 units of the same kind or quality, must leave 3 units of the same kind or quality. That is,

But these remainders are the same as we shall obtain by changing the signs of the subtrahends and then adding the results, algebraically, to the minuends. Thus,

+8a-(+5a)=+8a-5a=+3a
-8a-(-5a)=-8a+5a=-3a

Hence, in Algebra,

Subtracting any quantity consists in adding the same quantity with its sign changed.

54. This principle may be established in a more general manner as follows:

Let it be required to subtract the quantity b-c from a.

OPERATION.

Minuend,
Subtrahend,

Difference,

α

b-c

a-b+c

We first subtract b from a, indicating the operation, and obtain for a result, a-b. But the true subtrahend is not b, but b-c; and, as we have subtracted a quantity too great by c, the remainder thus obtained must be too small by c; we therefore add c to the first result, and obtain the true remainder, a-b+c. But this result is the same as would be obtained by adding —b+c to u. 55. It follows from the principle enunciated above, that any quantity is subtracted from nothing or zero, by simply changing its sign or signs. Thus,

56. From these principles and illustrations we deduce the following

RULE. I. Write the subtrahend underneath the minuend, placing the similar terms together in the same column.

II. Conceive the signs of the subtrahend to be changed, unite the similar terms as in addition, and bring down all the remaining terms with their proper signs.

12. From 8x-3xy+2y+c subtract x2-6xy+3y2-2c.

Ans. 7x+3xy-y2+3c.

13. From a+b subtract a—b.

Ans. 2b.

14. From xy subtract x-ly.

Ans. y.

15. From a+b+c subtract—a—b—c. Ans. 2a+2b+2c.

16. From 3a-b-2x+7 take 8—3b+a+4x.

Ans. 2a+2b-6x—1.

17. From 6y-2y-5 take -8y'-5y+12.

Ans. 14y+3y-17.