merely write the quantities after each other, without any change in the signs.

As another example, let it be required to add the polynomials a2+3ab+3b2-c2, and 5a2-7 ab+10b2 6 c2. The sum is a +3 ab+3 b2 — c2 + 5 a2 — 7ab10b2-6 c2; but this result contains similar terms, and may be reduced; it then becomes 6 a2-4 ab+13b2 -7 c2, which is the sum in its simplest form.

ART. 36. From what precedes, we have the following

RULE FOR THE ADDITION OF ALGEBRAIC QUANTITIE

Write the several quantities one after another, givirg to each term its original sign, and then reduce the similar terms.

Remember that those terms which have no sign, ar❤ supposed to have the sign +.

1. Add 3 a, 7b, 6 c, and 4 a +26.

2. Add 2 a2, 4 m2+3 m n, and 7 a2+6m n.

3. Add 4 a, 3a-2 c, and 5m+7c.

4. Add a2+2ab+b2, and 3 a2-2ab+4b2.

5. Add 3x2-15 x y + 11 y2, 15 x y, and 20 x2-13 y +m2.

6. Add 4 c5 m2, 3m2-2 c2, 15 c2-10 m2, and x +3y2-c2.

7. Add 20+3x2, 17x2+5y2, 45-7x2, and -3 y2.

8. Add 12 a2x-20 a x2+8x3, and 24 a3 +40 a x2

9. Add a + a1 c2 + a2 c4, and c6 — aa c2 — a2 c1.

10. Add 233 a x2+4a2 x+2, and 4x3-3 a x2+ 10 a2 x — 1.

-

b2

11. Add 8a2b-5ab2-8abc+4bc2, and 6 a b22a2b-abc-4bc2.

12. Add 6ab2ac-3bc, 4bd-7ab, and 6bc +5bd-3 ac.

13. Add 10+x2, 30-3 y2, 4a2b2+3 c2, 11 x2+ 7y2, and 23-a2 b2.

14. Add 17 a b2 - 10 a b2 c +5 a b c2, 45 a2 b2 — 10 a b c2, and 25+ 10 a 62 c.

15. Add a3 pm + a p m3, 7a3 pm + a3 mp+6 am p2, -12 a3 m p.

6+3 a p2 m-4 a p m3, and 10 a3 m p-8 am p2

16. Add a2b2-25x2, 3 a2 b2 + y2-4 x2, 759a2 b2+3y2, and 45 x2 y2 + 3 x2 — 9.

SECTION X.

SUBTRACTION.

ART. 37. We have already seen that the subtraction of a positive monomial consists in giving it the sign, and writing it either after or before the quantity from which it is to be subtracted. Thus, to subtract b from a, we write a-b, or -b+a.

If we have to subtract a polynomial in which all the terms are affected with the sign +, it is plain that each of the terms must be subtracted, that is, the sign of each term must be changed to —.

As an example in figures, if it is required to subtract 7+3 from 18, we must subtract both 7 and 3; thus, 18 -7-3, which reduced becomes 8. This result is correct, because 7+ 3 is 10, and 10 taken from 18 leaves 8.

In like manner, to subtract m+n from a, we write a― m-n, changing the signs of both m and n.

If some of the terms in the quantity to be subtracted the signs of those terms must be

have the sign changed to +.

As an example in numbers, let us subtract 7-4 from 12; 7—4 is 3, and 3 subtracted from 12 leaves 9. Now, to subtract 7 — 4 without reducing it, if we first subtract 7, which is expressed thus, 12-7, we subtract too much by 4, and the remainder, 12-7, or 5, is too small by 4; consequently, after having subtracted 7, we must add 4, which gives 12-7+4, or 9, for the true remainder.

In like manner, a-b subtracted from c, gives c—a +b. For, if a be subtracted from c, we have c—a; but since a is greater than a―b by b, we have subtracted too much by b, and the remainder, c— a, is too small by b; we must therefore add b to c -a, and we have cab for

the true result.

For another example, let us subtract a2-2ab+3 m2 from 5a2+4 ab-2m2. As in the previous examples, it may be shown that the terms in the quantity to be subtracted must have their signs changed. Making this change in the signs, and then writing the quantities after each other, we have, for the remainder, 5 a2 + 4 a b − 2 m2 — a2+2ab-3 m2. This remainder is correct, but it may be reduced, and it then becomes 4a2+6 ab— 5 m2, which is the remainder in its simplest form.

ART. 38. From what precedes, we derive the following

RULE FOR THE SUBTRACTION OF ALGEBRAIC QUANTITIES.

Change the signs of all the terms in the quantity to be subtracted, from + to or from to +, and write it

after that from which it is to be subtracted; then reduce similar terms

1. From 11 a2b—6 a b2, subtract 4 a2b-7 u b2 + m2. Changing the signs of all the terms in the latter quantity, and writing the result after the former, we have 11 a2b-6 a b2 — 4 a2 b + 7 a b2 m2, which, reduced, becomes 7a2bab2-m2, Ans.

2. From a2+2ab+b2, subtract a2-2ab+b2.

3. From 3 m2+9mc+2c2, subtract 3 m2-9mc + 2 c2.

4. From 6 a2b2+17 a b c, subtract 3 a2 b2-5 abc. 5. From 25+x, subtract 25-3 x.

6. From 22+27, subtract x2 — 27. 7. From 10 a2 b2 c · +6a2b2c-3 m2.

-13 m n +4 m2, subtract - 15 m n

-

8. From 8ab2cd+5ac-7ad, subtract 3 ab+ 4cd+5 ac.

9. From 3bd+2a+m, subtract 2bd-3 a-b. 10. From 7a2 b2c+5 a b2 c2-9abc+m, subtract 2 a2 b2c-4ab2 c2-8abc-2m.

ART. 39. The subtraction of a polynomial is indicated by enclosing it within brackets, or a parenthesis, and placing the sign before the whole. Thus, 3 m— [a2-7ab+m], or 3 m — (a2-7 ab+m), indicates the subtraction of a2-7 ab+m from 3m. Performing this subtraction, recollecting that a2 is affected with the sign +, we have 2 m - a2 +7 ab.

Let the learner perform the subtraction indicated in the following examples.

1. 3a2+b-(a2-10b+c2).

2. 4 m2-n2-(7 ab-6 m2+3 n3).

3. a2+2ac+c2 (a2 2 ac+c2).

4. 5abc+14 m2 - (10 abc-3 m2—3 cd).

5.-7 a x2 y2+3x2 y―(15 x2y-10 x2 y2-5abc). 6. a2+6ax +10 a x2-(a2-6 a x — 10 a x2).

7. m1- (a2-2 m1). m4.

8. 7xz-3x22+3z3-(3 x 22-7xz+10 23).

9. 4x2 y2-4 x y3 — 1 — (2 x2 y2—4 x y3 +5).

—

10 30 ab- 15 ac2 + 15 m2 x − (15 a c2 + 30 ab — 15 m2 x).

ART. 40. It is often useful to reverse the process in the last article, and place part of a polynomial within a parenthesis, preceded by the sign —. This may be done without altering the value of the polynomial, provided the signs of all the terms placed within the parenthesis are changed. Thus, a -m+n may be written a― (m—n); for if the subtraction indicated in the latter expression be performed, we obtain a -m+n.

Let the student throw all, except the first term, of each of the following polynomials into a parenthesis, preceded by the sign

1. m2-ca.

2. 4a2b-3 c2.

3. 150-2x+4y-7ab.

4. 6x2 y2+2 a x + 3by-10 z2.
5. a2 m2-10+6xy.
a2

6. 3abc+mx-3y+10.

7. 4x2 y3-6x y3+ abc-m2 x.

8. 20 p2q-56+x2.

9. a2 b2c2+10-a3bc2abc.