NOTE. The learner should be reminded, that the quantities in the parentheses are to be considered as one quantity; then it is evident, that 3 times, 5 times, and 7 times any quantity whatever, will be equal to 15 times that quantity.

Add together

8. 3a(b+x), 5a(b+x), 7a(b+x), and -11a(b+x).

Ans. 4a(b+x).

9. 2c(a2-b2), -3c(a2—b2), 6c(a2—b2), and —4c(a2-b2).

Ans. c(a-b2).

10. 3az-4by-8, -2az+5by+6, 5az+6by-7, and -8az -7by+5. Ans. -2az-4.

11. 8ax-3cz2,—5ax+5cz2, ax+2cz2, and −4ax-4cz2. Ans. 0. 12. 8a+b, 2a-b+c, -3a+5b+2d, -6b-3c+3d, and -5a +7c-2d. Ans. 2a-b+5c+3d.

13. 7x-6y+5z+3-g,-x-3y-8-g,-x+y-32-1+7g,-2x +3y+3z—1—g, and x+8y—5z+9+g. Ans. 4c+3y+2+5g. 14. 2a2+5ab-xy, -7a2+3ab-3xy, -3a2-7ab+5xy, and 9a2 -ab-2xy. Ans. a2-xy.

15. 5ab2-8a2b3+x2y+xy2; 4a2b3 — 7a3b2—3xy2+6x2y, 3a3b2 +3a2b3-3x2y+5xy2, and 2a2b-a3b2-3x2y-3xy2. Ans. a2b3+x2y.

SUBTRACTION.

ART. 56. SUBTRACTION in Algebra, is the process of finding the simplest expression for the difference between two algebraic (quantities.

In Algebra, as in Arithmetic, the quantity to be subtracted is called the subtrahend. The quantity from which the subtraction is to be made, is called the minuend. The quantity left, after the subtraction is performed, is called the difference, or remainder.

REMARK. The word subtrahend means, to be subtracted; the word minuend, to be diminished.

1. Thomas has 5a cents; if he give 2a cents to his brother, how many will he have left?

Since 5 times any quantity, diminished by 2 times the same quantity, leaves 3 times the quantity, the answer is evidently 3a; that is 5a-2a-3a.

Hence, to find the difference between two positive similar quantities, we find the difference between their coefficients, and prefix it to the common letter, or letters.

Let it be noted, that the sign of the quantity to be subtracted, is changed from plus to minus.

Ans. 362xy.

12. From 762xy, take 4b2xy. ART. 57.-1. Thomas has a number of apples, represented by a; if he give away a quantity, represented by b, what expression will represent the number of apples he has left?

If a represents 6, and b 4, then the number left would be represented by 6-4, which is equal to 2; and whatever numbers a and b represent, it is evident that their difference may be expressed in the same way, that is, by a-b.

Hence, to find the difference between two quantities that are not similar, we place the sign minus before the quantity that is to be subtracted.

Let the pupil here notice again, that the sign of the quantity to be subtracted, is changed from plus to minus.

ART. 58.-1. Let it be required to subtract 5+3 from 9.

If we subtract 5 from 9, the remainder will be 9-5; but we wish to subtract, not only 5, but also 3; hence, after we have subtracted 5, we must also subtract 3; this gives for the remainder, 9-5-3, which is equal to 1.

REVIEW.-56. What is Subtraction in Algebra? What is the quantity to be subtracted, called? What is the quantity called, from which the subtraction is to be made? What does subtrahend mean? What does minuend mean? How do you find the difference between two positive similar quantities? 57. How do you find the difference between two quantities that are not similar?

2. Again, suppose that it is required to subtract 5-3 from 9. If we subtract 5 from 9, the remainder will be 9-5; but the quantity to be subtracted is 3 less than 5, and we have, therefore, subtracted 3 too much; we must, therefore, add 3 to 9-5, which gives for the true remainder, 9—5+3, which is equal to 7.

3. Let it now be required to subtract b―c from a.

If we take b from a, the remainder is a-b; but, in doing this, we have subtracted c too much; hence, to obtain the true result, we must add c. This gives, for the true remainder, a—b+c. If a=9, b=5, and c=3, the operation and illustration by figures would stand thus: from a

take b-c

Remainder, a-b+c

from 9

take 5-3

Rem. 9-5+3

=9

The same principle may be further illustrated by the following examples.

4. a-(c-a)=a-c+a=2a-c.

a—(a—c) —a—a+c=c.

a+b—(a—b) =a+b—a+b=26.

Let it be noted, that in the result in each of the preceding examples, the signs of the quantity to be subtracted have been changed from plus to minus, and from minus to plus; hence, in order to subtract a quantity, it is merely necessary to change the signs and add it. Hence, the

RULE,

FOR FINDING THE DIFFERENCE BETWEEN TWO ALGEBRAIC QUANTITIES.

Write the quantity to be subtracted under that from which it is to be taken, placing similar terms under each other. Conceive the signs of all the terms of the subtrahend to be changed, and then reduce the result to its simplest form.

NOTE. It is a good plan with beginners, to direct them to write the example a second time, and then actually change the signs, and add, as in the following example. They should do this, however, only till they become familiar with the rule.

From 5a+3b-c

The same, with the
Take 2a+2b+3c signs of the subtra-

Remain. 3a+56+2c hend changed.

5a+3b-c
2a+2b+3c

3a+5b+2c

12. From 14, take ab-5.
13. From a+b, take a.
14. From a, take a+b.
15. From x, take x-5.
16. From 3ax, take 2ax+7.
17. From x+y, take x―y.
18. From x-y, take x+y.
19. From x- -y, take y—x.
20. From x+y+z, take x-y-z.
21. From 5x+3y-z, take 4x+3y+z.
22. From a, take -a.
23. From 8a, take -3a.
24. From a, take -4a.

.

25. From 5b, take 11b.
26. From a, take -b..
27. From 3a, take -2b.
28. From -9a, take 3a.
29. From -7a, take -7a.
30. From 19a, take -20a.
31. From 6a, take -5a..
32. From -3a, take -5b.
33. From -13, take 3.
34. From -9, take -16.
35. From 12, take -8.
36. From -14, take -5.

Ans. -3a+5b, or 5b-3a.

Ans. -16.

37. From 3a-2b+6, take 2a-7b-3.

38. From 13a-2b+9c-3d, take 8a-6b+9c-10d+12.

Ans. 5a+4b+7d—12.

39. From -7a+3m-8x, take -6a-5m-2x+3d.

44. From 2x2-3a2x2+9, take x2+5a2x2-3. Ans. x2-8a2x2-12. 45. From 4x2y3-5cz+8m, take —cz+2x2y3—4cz.

Ans. 2 x2y+8m.

46. From 2-11xyz+3a, take -6xyz+7-2a-5xyz.

48. From 3a(x-z), take a(x—z). .

Ans. 2a (x-2).

49. From 7a (c-z)-ab(c-d), take 5a2(c-z)-5ab(c-d). Ans. 2a (c-z)+4ab(c-d).

ART. 59. It is sometimes convenient to indicate the subtraction of a polynomial without actually performing the operation. This may be done, if it is a monomial, by placing the sign minus before it; and, if it is a polynomial, by enclosing it in a parenthesis, and then placing the sign minus before it.

Thus, to subtract a-b from 2a, we may write it 2a—(a—b), which reduces to a+b.

By this transformation, the same polynomial may be written in several different forms; thus:

a-b+c-d-a-b―(d—c)=a-d—(b—c)—a—(b—c+d)

Let the pupil, in each of the following examples, introduce all the quantities, except the first, into a parenthesis, and precede it by the sign minus, without altering the value of the expression. 1. a-b+c.

Ans. a-(b-c).

Ans. b-(d-c).

Ans. 2-(2xy-z).

Ans. ax-(cd-be-h).

5. m-n-2-8.

6. m―n+z+8.

Ans. m

It will be found a useful exercise for the pupil, to take each of the preceding polynomials, and without changing their values, write them in all possible modes, by including either two or more terms in a parenthesis.

OBSERVATIONS ON ADDITION AND SUBTRACTION.

ART. 60. It has been shown, that Algebraic Addition is the process of collecting, into one, the quantities contained in two or more expressions. The pupil has already learned, that these expressions may be all positive, or all negative, or partly positive and partly negative. If they are either all positive, or all negative, the sum will be greater than either of the individual quantities; but, if some of the quantities are positive and others negative, the aggregate may be less than either of them, or, it may even be

REVIEW. In subtracting b-c from a, after taking away b, have we Bubtracted too much, or too little? What must be added, to obtain the true result? Why? What is the general rale for finding the difference between two algebraic quantities? 59. How can the rubtraction of an algebraic quantity be indicated?