66. Hence, we have the following general rule for the subtraction of algebraic quantities.

RULE.

Change the signs of all the quantities to be subtracted into the contrary signs, or conceive them to be so changed, and then add, or connect them together, as in the several cases of addition.

EXAMPLE 1. From 18ab subtract 14ab.

Here, changing the sign of 14ab, it becomes -14ab, which being connected to 18ab with its proper sign, we have 18ab-14ab (18-14)ab= 4ab. Ans.

Ex. 2. From 15x2 subtract -10x2.

Changing the sign of ---10, it becomes +10.c2, which being connected to 15% with its proper sign, we have 15x2+10x225x2. Aus.

Ex. 3. From 24ab+7cd subtract 18ab+7cd. Changing the signs of 18ab+7cd, we have -18ab -7ed, therefore, 24ab+7cd-18ab-7cd=6ab.

Ex. 4. Subtract 7a-5b+3ax from 12a+10b+

13ax-3ab.

Changing the signs of

all the terms of 7a-5b Sax; it becomes,

12a+106+13ax-3ab

-7a+5b-3ax S

.. by addition, 5a+156+10ax-3ab.

Ex. 5. From 3ab-7ax+7ab+Sax, take

4ab-3ax-4xy,

Changing the signs of all the terms of 4ab-3ax-4xy, S

Sab-7ax

7ab+3ax

--4ab+3ax+4xy

.. by addition, 6ab-ax+4xy. Ans.

In the above example, one row is set under the other, that is, the quantities to be subtracted in the lower line; then, beginning with 14a, and conceiving its sign to be changed, it becomes -14a, which being added to 36a, we have 36a-14a=22a; also. -46, with its sign changed, added to -126 will give 4b-12b-(4-12)b=-8b; in like manner, 7c-7e =0, and -8, with its sign changed, +8. The following examples are performed in the same manner as the last.

Ex. 7.

=

From 3x-4a+ b

Ex. 8. a + b

Rem. Sab-4ax+y--4cx--2x2+3y2 x3+5x2-9x

3

(b−α)x3+(q+c)x2 −(r+m)x+(p+s)y2

67. As quantities in a parenthesis, or under a vinculum, are considered as one quantity with respect to other symbols (Art. 10,) the sign prefixed to quantities in a parenthesis affects them all; when this sign is negative, the signs of all those quantities must be changed in putting them into the parenthesis.

Thus, in (Ex. 13), when cx is subtracted from -, the result is -bx+cx2, or (b-c)x2; because the sign prefixed to (b-c) changes the signs of b and c; or it may be written +(c-b). Again, in (Ex. 14), when +mx is subtracted from -ra, the result is -rx-mx; and, as this means that the sum of rx and mx is to be subtracted, that negative sum is to be expressed by-(rx+mx)=m)x. For the same reason, the multinomial quantity my+n2y2 —aby 3 —ry2+6y2, when puć

3

into a parenthesis, with a negative sign prefixed,

becomes

-(m—n2+ab+r-6)y2.

Ex. 15. From a-b, subtract a+b.

Ans. 26. Ex. 16. From 7xy-5y+3x, subtract 3xy+3y Ans. 4xy-8y.

+3x.

Ex. 17. What is the difference between 7ax2+ 5xy-12ay+5bc, and 4ax2+5xy-8ay-4cd.

Ans. 3ax2-4ay+5bc+4cd. Ex. 18. From 3x2-3ax +5, take 5x2+2ax+5. Ans. 3x2-5ax.

Ex. 19. From a+b+c, take-a-b-c.

Ans. 2a+26+2c.

Ex. 20. From the sum of 3x3-4ax+3y2, 4y2+ 5ax—x3, y2 —ax+5x3, and 3ax-2x2-y2; take the sum of 5y — x2+x3, ax-x3+4x2, 3x3-Ɑx · 3y2, and 7y2-ax+7.

3

Ans. 4x3+4ax-2y2 —5x2 —7. Ex. 21. From the sum of x3y2x2y-3xy3, 9xy-15-3x2 y2, and 70+2x2 y2-3x2y; subtract the sum of 5x3y2 —20+xy2, 3x2y-x3y2+ax, and 3xy2-4x2y2-9+a2x2.

Ans. 2xy2-7x2y-ax-a2x2+84. Ex. 22. From a3x3y2-m2x3+3cx-4x-9: take a2x2y2 —n2 x3 + c2x+bx2+3.

Ans. (a3-a2)x2y3 — (m2 —n2 )x3 +(3c →c2) xx-(1+b) x2-12.

§ III. Multiplication of Algebraic Quantities.

In the multiplication of algebraic quantities, the following propositions are necessary to be observed.

68. When several quantities are multiplied continually together, the product will be the same, in whatever order they are multiplied.

Thus, axb-bxa=ab.

For it is evident, from the nature of multiplication, that the product contains either of the factors as many times as the other contains an unit. Therefore, the product ab contains a as many times as contains an unit, that is, 6 times.

α

And the same quantity ab, contains b as many times as a contains an unit, that is, a times. Consequently, axb=ba=ab; so that, for instance, if the numeral value of a be 12, and of b, 8, the product ab, will be 12 x 8, or 8 X 12, which, in either case, is 96.

In like manner it will appear that abc=cab= bca, &c.

69. If any number of quantities be multiplied continually together, and any other number of quantities be also multiplied continually together, and then those two products be multiplied together; the whole product thence arising will be equal to that arising from the continual multiplication of all the single quantities.

Thus, ab Xcd=axbXcxd=abcd.

For aba×b, and cd=cXd; if x be put =cd, then ab X cd abxx axbxx ; but x is=cd=cXd. .ab Xx=ab×c × d=axbxcd=abcd.

·70. If two quantities be multiplied together, the pro duct will be expressed by the product of their numeral coefficients with the several letters subjoined. Thus, 7a X 5b=35ab.

For 7a is 7 Xa, and 56=5xb, ...7aX5b7Xα X5×6=7× 5 ×a×b=35 Xab≈35ab.