Hence a3b2 ÷ a3b = a3 × b2 ÷ a3 ÷ b = a5 ÷ a2 × b2 ÷ b = a3b,

and

a2bca2b3c* = ab3.

36. We have proved in Art. 28 that

ax (-b) = -ab;

:: (−ab)÷(-b) = a, and (— ab) ÷ a = − b ;

we have also proved that

(a) (b) = +ab = (+ a) (+ b) ;

:: (+ab)÷(− a) = − b, and (+ ab) ÷ (+ a) = + b.

Hence if the signs of the dividend and divisor are alike, the sign of the quotient is +; and if the signs of the dividend and divisor are unlike, the sign of the quotient is; we therefore have the same Law of Signs in division as in multiplication.

1. Divide 10a by - 2a, 3a2b3 by - 2ab3, and -7a5b3c4 by - 3a2b2c2.

2. Divide -2a5b7c8 by 4a3bc7, - 6x3у1 by 3x3y, and -5a2b1xy by

- 2ab4x2y5.

Ans. -1a2bc, -2x2y3, 5 ax3y3.

2

3. Multiply-2a3bc5 by - 3ab7c2 and divide the result by 8a3b6c®.

3 Ans. ab2c. 4

37. The fundamental laws of Algebra, so far as monomial expressions are concerned, are those which were

marked A, B, C, D in the preceding articles, and which are collected below:

It should be remarked that the laws expressed in (A), (B), (C) have been proved to be true for all values of a and b; but both m and n are supposed in (D) to be positive integers.

MULTINOMIAL EXPRESSIONS.

38. We now proceed to the consideration of multinomial expressions.

We first observe that any multinomial expression can be put in the form

a+b+c+&c.,

where a, b, c, &c., may be any quantities, positive or negative.

For example, the expression 3x3y - §xy2 - 7xyz, which by (A) is the same as 3xy+(— § xy3) + (− 7xyz), takes the required form if we put a for 3x2y, b for - §xy3, and c for (-7xyz).

It therefore follows that in order to prove any theorem to be true for any algebraical expression, it is only necessary

to prove it for the expression a+b+c+ &c., where a, b, c, &c. are supposed to have any values, positive or negative.

39. It follows at once from the meaning of addition that the sum of two or more algebraical quantities is the same in whatever order they are added. For example, to find how much a man is worth, we can take the different items of property, considering debts as negative, in any order.

Thus a+b+c=c+a+b=b+c+ a = &c. ......(E).

The laws [C] and [E] are together called the Commutative Law, which may be enunciated in the following form: Additions or Multiplications may be made in any order.

40. Since additions may be made in any order, we have a + (b+c+d+ ...) = (b + c + d + ...) + a (from E)

=a+b+c+d+... (from E).

Hence, to add any algebraical expression as a whole is the same as to add its terms in succession.

Since the expression + ab + c d may be written in the form +a+(− b) + c + (− d), we have

+{+ab+c-d} = + {+ a + (− b) + c + (− d)}

=+a+(-b)+c+ (−d).

When we say that we can add the terms of an expression in succession, it must be borne in mind that the terms include the prefixed signs.

41. Since subtraction and addition are inverse operations, it follows from the preceding that to subtract an expression as a whole is the same as to subtract the terms in succession. Thus

a-(b+c+d+...) a-b-cd-...

=

42. If c be any positive integer, a and b having any values whatever, then

...

(a + b) c = (a + b) + (a + b) + (a + b) + repeated c times = a+b+a+b+a+b+... [Art. 40]

Hence, when c is a positive integer, we have

(a + b) c = ac + bc.........

(F).

Since division is the inverse of multiplication, it follows that when d is any positive integer

Thus the law expressed in (F) is true for all positive values of c; and being true for any positive value of c, it must also be true for any negative value. For, if

Hence for all values of a, b and c we have

(a + b) c = ac + bc.............

Thus the product of the sum of any two algebraical quantities by a third is the sum of the products obtained by multiplying the quantities separately by the third.

The above is generally called the Distributive Law.

43. Since

(a + b) + c = (a + b) × 2/

1

1

= ax + b x = a + c + b + c,

we see that the quotient obtained by dividing the sum of any two algebraical quantities by a third is the sum of the quotients obtained by dividing the quantities separately by the third.

44. From Art. 40 it follows that

a+b+c+d+e+... = (a+b)+c+d+e) +...

= a+ (b+c+d) + e +... = &c., so that the terms of an expression may be grouped in any

manner.

Again, from Art. 29, it follows that

abcde... =a (bc) (de).. = a (bcd) e

...

so that the factors of a product may be grouped in any

manner.

These two results are called the Associative Law.

45. We have now considered all the fundamental laws of Algebra, and in the succeeding chapters we have only to develope the consequences of these laws.