EXAMPLES FOR PRACTICE.

61. Remove the parentheses from the following expressions, and

reduce the results:

1. 3a+(262-a-d+m).

2. 4x3-y—(3x-7y+5)+2x.

3. a+2c-(4c-3a+2m3).

Ans. 2a+26-d+m.

Ans. 4x+6y-x-5.

4. 4x3—2x2-[xa—(2x2+5x—7)—6x+1].

5. a+2m—{c+x—[a—m—(c—2x)]}·

Ans. 4a-2c-2m3.

Ans. 3x+11x-8.

Ans. 2a+m-2c+x.

·6. 3x2—4x—am—{:
7. 3a—{2m2+[5c—9a—(3a+m3)]+6a—(m2+5c)};

-X- —[3am—(2x+2am)+2x3]—5am}.
Ans. 4x-5x+5am.

Ans. 9a.

8. x2—{5mc2—[x2—(3c—3mc3)+3c—(x2—2me3—c)]}·

Ans. x2+c.

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

10. 5%3-3x2+4%-1-[2-(3≈-2z+1)-2*+*]·

Ans. 2m+2.

Ans. 42+z.

11. 4c3—2c2+c+1—(3c3—c3—c—7)—(c3—4c2+2c+8).

Ans. 3c'.

12. 3ab4cd-(3cd—2a1b) — [ a'+c— (5cd+3ab)+(3a2 +2cd)+a'].

13.

Ans. 8a2b-4cd-5a3-c.

· ( 4a23m+3m3d—(7m3d—9a3m—n)— { 5n—[m3d—(2n+

am)+3ax"]—ba'm)—12am).

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

62. In Algebra, addition does not necessarily imply augmentation, nor does subtraction always imply diminution, in an arithmetical 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 b 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 36. Since it is immaterial in what order the factors are taken, we may proceed thus: 5X3=15; axbab; and 15Xab=15ab. Or 5a3b=15ab. Hence,

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

2d. The law of exponents.

Let it be required to multiply ab3 by a3b'. Since a*b*= aaaa bbb, and a3b'=aaa bb, we have

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

subtractively, or —a—a—a— etc., to d repetitions; and this is the product of a by-d. Hence we conclude that the signs, + and -, when prefixed to a multiplier, must be interpreted as follows:

The plus sign before a multiplier shows that the multiplicand is to be successively added; and the minus sign before a multiplier shows that the multiplicand is to be successively subtracted.

To exhibit the law which governs the sign of a product, according to this principle, we present the four cases which involve all the variations of signs. It will be observed that according to the above interpretation, the multiplicand is to be repeated with its proper sign when the multiplier is positive, but with its sign changed when the multiplier is negative. We shall therefore have the fol lowing results:

1.

2.

3.

4.

+ax(+b)=+a+a+a+etc.=+ab.

+ax(-b)=-a-a-a-etc.-ab.

-ax(+b)=-a-a-a-etc.-ab.

—a×(—b)=+a+a+a+etc.=+ab.

Comparing the first result with the fourth, and the second with the third, we observe that

When the two factors have like signs, the product is positive; and when the two factors have unlike signs, the product is negative.

67. This law applied in the case of three or more negative factors gives the following results :

= +ab

=(+ab )X(-c )——abc =(-abc)X(-d)=abcd

(-a)X(-b)X(-—c)X(—d)X(-e)=(+abcd)X(-e)=abcde

Hence the general truth:

The product of an even number of negative factors is positive; and the product of an odd number of negative factors is negative.

CASE I.

68. When both factors are monomials.

From the principles already established we derive the following RULE. I. Multiply the coefficients of the two terms together for the coefficient of the product.

II. Write all the letters of both terms for the literal part, giving each an exponent equal to the sum of its exponents in the two terms. III. If the signs of the two terms are alike, prefix the plus sign to the product; if unlike, prefix the minus sign.

15. What is the continued product of 3x, 2x'y, and 7x3y3z?

Ans. 42xyz.

16. What is the continued product of 5a3b, ab3, 3a2c, and -5abc? Ans. -75a b°c2.

17. What is the continued product of 7xy, -2x3, 3x1y, —xy', and xy'?

18. What is the continued product of -3c'dm, -2cdm, and --5cdm2? Ans. -30c'd*m*. 19. What is the continued product of -a, -ab, —abc, -abcd. -abcdh, and -abcdhm?

20. Multiply 2(x+y) by 4a2(x+y). 21. Multiply 4m2(x-2)' by (≈—x). 22. Multiply (a–c)m+1 by (a-c)TM-1.

Ans. a b'c'd h3m.

Ans. 8a'(x+y)'. Ans. -4m2(x-z)3. Ans. (a-c).