first term. Passing then to the terms involving 62, we find their sum to be - 562, after which we write - 3 c2.

NOTE. The marks are drawn across the terms, that none of them may be overlooked or omitted.

5. If a= 5,

b

=

4, c = = 2, x = 1, what are the numerical

values of the several sums above found?

9a+f -6a+g

- 2 a

7x+3ab+3c
3x-3ab5c

5x-9ab-9c

8x2+ 9 acx + 13 a2b2c2 ---722-13 acx+14a2b2c2 4x2+ 4 acx 20 a2b2c2

7x-9y+5z+3- g 14.

x+y-3z+1+7g

19 ah2+3 a3b1 — 8 ax3 -17 ah2-9 a3b1+9ax3

8a+ b

x-3y

-8- g

-2x+6y+3z-1- g

-66-3c+3d

15. Add

b+3c-d-115e+6f-5g,

36 2c-3d-e+27f, 5c-8d+3f-7g,

-76-6c17d9e-5f+11g,
-36-5d-2e+6-9g+h.

Ans. 86-109 e +37f-10g+h.

16. Add the polynomials 7ab3abc-8b2c-9c3 + cơ2,

8abc-5a2b+3c46c+cd2,

and 4ab8c3+9bc-3d3.

Ans. 6ab5abc-3b2c - 14c3+2cď2 - 3 ď3.

17. What is the sum of 5 a2bc +6 bx − 4 af,
-3a2bc-6 bx+14 af, - af +9bx+2a2bc,
+6af8bx+6a2bc?

Ans. 10a2bc + bx + 15 af.

18. What is the sum of a2n2 + 3 a3m + b,

- 6a2n2 — 6 a3m — b, + 9b - 9 a3m — 5a3n2?

19. What is the sum of 4a3b'c - 16 a1x — 9 ax3d, +6a3b3c-6ax3d+17 a'x, +16 ax3d-a'x-9a3b2c?

Ans. a3b2c+ax3d.

20. What is the sum of 7g +36 +4g-2b+3g-3b

+26?

21. What is the sum of ab + 3xy - m―n,

Ans. 0.

-6xy-3m+11n+cd, +3xy + 4m-10n+fg? Ans. abcd+fg.

22. What is the sum of 4xy+n+6ax+9am, - 6xy+6n6ax-8am, 2xy - 7n+ax-am?

7(a+b)

NOTE.

6 (c3 — aƒ3)

The quantity within the parentheses must be taken as a whole (827). In Exercise 23 the sum of a and b, indicated by (a + b), is the unit; in Exercise 24 the difference of a2 and c2, indicated by (a2c2), is the unit.

26. Add 3a (g2 — h2) — 2 a (g2 — h3) + 4a (g2 — h3)

+8a (g2 — h2) — 2a (g3 — h2).

Ans. 11a (g2 - h2).

27. Add 3c (a'c — b2) — 9c (a2c — b2) — 7 c (a2c — b2)

-

-

+15c (a2c — b2)+ c (a2c — b2).

-

Ans. 3c (a'c-b2).

34. In algebra the term "add" does not always, as in arithmetic, convey the idea of augmentation; nor the term "sum" the idea of a number numerically greater than any of the numbers added: for, if to a we add b, we have a b, which is, arithmetically speaking, a difference between the number of units expressed by a and the number of units expressed by b; consequently this result is numerically less than a. To distinguish this sum from an arithmetical sum, it is called the algebraic sum.

SUBTRACTION.

35. Subtraction is the operation of finding the difference between two algebraic quantities.

36. The subtrahend is the quantity to be subtracted; the minuend, the quantity from which it is taken.

37. The difference of two quantities is such a quantity as, added to the subtrahend, will give a sum equal to the minuend. (1) From 17 a take 6a.

In this example, 17a is the minuend, and 6a the subtrahend. The difference is 11 a, because 11 a added to 6a gives

17 a

6 a

17 a.

11 a

NOTE. The difference may be expressed by writing the quantities thus: 17a - 6 a 11 a, in which the sign of the subtrahend is changed from + to

(2) From 15x take - 9x.

The difference, or remainder, is such a quantity as, being added to the subtrahend (— 9x), will give the minuend (15x). That quantity is 24 x, and may be found by simply changing the sign of the subtrahend, and adding: whence we may write, 15x-(-9x) = 24 x.

15x

9x

24x

(3) From 10ax take a - b.

10ax

The difference, or remainder, is such a quantity as, added to a – b, will give the minuend (10 ax). What is that quantity? If you change the signs of both terms of the subtrahend, and add, you have 10 ax -a+b. Is this the true remainder? Certainly for, if you add the remainder to the subtrahend (a - b), you obtain the minuend (10 ax).

:

+a-b 10ax-a+b

10 ax

+a-b

It is plain, that if you change the signs of all the terms of the subtrahend, and then add them to the minuend, and to this result add the given subtrahend, the last sum can be no other than the given minuend: hence the first result is the true difference, or remainder (8 37).

From the preceding examples we have, for the subtraction of algebraic quantities, the following rule:

Write the terms of the subtrahend under those of the minuend, placing similar terms in the same column.

Conceive the signs of all the terms of the subtrahend to be changed from + to or from to +, and then proceed as in addition.

NOTE. Exercise 2 is the same as Exercise 1, with the signs of the subtrahend changed.