18. From 3p+q+r-3s take q—8r+2s—8.

Ans. 3p+9r-58+8.

19. From 13a-2ax+9x1 take 5a3-7ax-x3.

Ans. 8a +5ax+10x3.

20. From x-3x+5x-7x+12 take x-4x+2x2-6x+15.

Ans. x+3x-x-3.

21. From a3a'c+5a'c'-2a2c+4ac-c take a°-4a'c+ 2a'c-5a2c+3ac-c. Ans. a'c+3a'c'+3a'c'+ac*.

22. From 2x+28x+134x-252x+144 take 2x+21x+ 67x2-63x+84. Ans. 7x+67x2-189x+60. 23. From x+5x*y+10x3y2+10x3y3+5xy*+y0 take xo—5x*y+ 10x3y2-10x'y'+5xy*—y*. Ans. 10x'y+20x*y*+2y*

24. From the sum of 6x3y-11ax' and 8x3y+3ax3, take 4x3y— 4ax+a. Ans. 10x'y-4ax3-a. 25. From the sum of 8cdx+15a'b-3 and 2cdx-8a2b+24 take the sum of 12a3b-3cdx-8 and cdx-4a3b+16.

Ans. 12cdx-ab+13.

57. The difference of two dissimilar terms may often be conveniently expressed in a single term, as in (51), by taking some common letter or letters as the unit of subtraction.

Ans. (am)x+(b—n)y+(c—p)z.
Ans. (c-1)x.

6. From ax+bx+cx take x+ax+bx.
7. From (a+2b+c) √xy take (26—c)√xу.

Ans. (a+2c)√xy.

8. From (3a-2m)x+(5a+2m) x2+(4am)x take (a-m)x -(2a+m)x2+(2a−3m)x.

Ans. (2a-m)x+(7a+3m)x2+(2a+2m)x.

9. From 1+2az*+·3a*z*+4a3z°+5a*z* take z3+2az*+3a3z°+ 4a*z*.

Ans. 1+(2a-1)2+(3a2—2a)z*+(4a'—3a*)z*+(5a'—4a3)z".

USE OF THE PARENTHESIS.

58. The term, parenthesis, will be employed hereafter as a general name to designate the various signs of aggregation employed in algebraic operations. The following rules respecting the use of the parenthesis should be thoroughly considered by the learner, if he would acquire facility in algebraic transformations.

59. From the definition of the signs of aggregation, (17), we understand that if the plus sign occurs before a parenthesis, all the terms inclosed are to be added, which does not require that the signs of the terms be changed; but if the minus sign occurs before a parenthesis, all the terms inclosed are to be subtracted, which requires that the signs of all the terms be changed. Hence,

1. A parenthesis preceded by the plus sign may be removed, and the inclosed terms written with their proper signs. Thus,

a―b+(c-d+e)=a—b+c-d+e

2. Conversely: Any number of terms, with their proper signs, may be inclosed by a parenthesis, and the plus sign written before the whole.

Thus,

a-b+c―d+e=a+(—b+c―d+e)

3. A parenthesis preceded by the minus sign may be removed, provided the signs of all the inclosed terms be changed. Thus, a―(b-c+d—e)=a—b+c―d+e

4. Conversely: Any number of terms may be inclosed by a parenthesis, preceded by the minus sign, provided the signs of all the given terms be changed. Thus,

a―b+c―d+e-a-b+c—(de)

60. When two or more parentheses are used in the same expression, they may be removed successively by the above rules. Thus, a- — { b—c—(d—e)}=a—{b—c―d+e}—a—b+c+d—e

Or, in a different order,

--{ b—c—(d—e) }—a—b+c+(d—e)=a−b+c+d—e

EXAMPLES FOR PRACTICE.

61. Remove the parentheses from the following expressions, and

6. 3x2—4x—am—{ x2-x-[3am—(2x+2am)+2x3]—5am}.

Ans. 4x-5x+5am.

—{2m2+[5c—9a—(3a+m3)]+6a—(m2+5c)}.

Ans. 9a.

8. x2—{5mc2—[x2—(3c—3mc3)+3c—(x2—2mc3—c)]}·

Ans. x2+c.

9. m3—m—1—{m3—2m—2—[m2—3m—3—(m2—4m—4)]}.

10. 5%3-3x2+4z—1—[2x3—(3≈3—2x+1)—z3+≈].

Ans. 2m+2.

Ans. 42+z.

11 4c-2c+c+1-(3c'—c2—c—7)—(c2—4c2+2c+8).

Ans. 3c.

12. 3ab4cd — (3cd—2a3b) — [ a3+c— (5cd+3a3b) +(3a2 +2cd)+a3]. Ans. 8a2b-4cd-5a3—c.

13.

( 4a3m+3m3d—(7m3d—9a3m—n)— { 5n—[m2d—(2n+ a2m)+3an"]—5am)—12am).

Ans. 3m'd+6n-5a3m-3an'.

62. In Algebra, addition does not necessarily imply augmentation, nor does subtraction always imply diminution, in an arithmet ical sense.

We have seen that one quantity is added to another by annexing it with its proper sign; but a quantity is subtracted from another by annexing it with its sign changed. Hence,

1st Adding a positive quantity has the same effect as subtracting a negative quantity; and adding a negative quantity has the same effect as subtracting a positive quantity.

2d. If to any given quantity a positive quantity be added, the result will be greater than the given quantity; but if a negative quantity be added, the result will be less than the given quantity.

3d. If from any given quantity a positive quantity be subtracted, the result will be less than the given quantity; but if a negative quantity be subtracted, the result will be greater than the given quantity.

63. Let -a denote any negative quantity. Add -b to this quantity, and subtract +b from it; and we have

—a+(-b)=—a—b

—a—(+b)=—a—b

But according to the last two propositions, the result, —a—b, should be less than the given quantity, -a. That is

-a-b <-a

Now, the quantity, -a-b, contains a greater number of units than -a. These cases, however, are not exceptions to the laws enunciated above; for in an algebraic sense, the less of two negative quantities is that one which contains the greater number of units. (See197).

64. If a represent the greater of the two numbers, and the less, then a+b is their sum and a-b their difference; and the sum and difference may be combined in two ways, as follows:

1. If the difference of two numbers be added to their sum, the

result will be twice the greater number.

2. If the difference of two numbers be subtracted from their sum, the result will be twice the less number.

MULTIPLICATION.

65. Multiplication, in Algebra, is the process of taking one quantity as many times as there are units in another.

66. In order to establish general rules for multiplication, we must first consider the simple case of multiplying one monomial by another; and we will investigate, first, The law of coefficients; second, The law of exponents; third, The law of signs.

1st. The law of coefficients.

Let it be required to multiply 5a by 3b. Since it is immaterial in what order the factors are taken, we may proceed thus: 5X3=15; axbab; and 15 Xab=15al. Or 5a3b=15ab. Hence,

The coefficient of the product is equal to the product of the coefficients of the multiplicand and multiplier.

2d. The law of exponents.

Let it be required to multiply ab3 by ab3. Since a*b*= aaaa bbb, and a3l2=aaa bb, we have

a'b'Xa'b' aaaabbbaaabb=a'b'.

=

Hence,

The exponent of any letter in the product is equal to the sum of the exponents of this letter in the multiplicand and multiplier.

3d. The law of signs.

In Arithmetic, multiplication is restricted to the simple process of repeating a number; and the only idea attached to a multiplier is, that it shows how many times the multiplicand is to be taken. In Algebra, however, a multiplier may be affected by either the plus or the minus sign; and it is necessary to consider how the sign of the multiplier modifies its signification.

For this purpose, suppose it were required to multiply any quantity, as a, by c-d. Now it is evident that a taken c minus d times, is the same as a taken c times, diminished by a taken d times ; or ax(c-d)=ac-ad. In the first term of this result, a is taken c times additively, or a+a+a+a etc., to c repetitions; and this is the product of a by +c. In the second term, a is taken d times