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ALGEBRA. 
same sign, the product is positive; if they have 
different signs, it is negative. 
1- 4*° X +5=4- a b ; because in this case 
a is to be taken positively b times; therefore 
the product a b must be positive. 
2. — a X -)-A = — ab; because — a is 
to be taben b times; that is, we must take 
— ab. 
3. -j- a. X — b — — ab ; for a quantity 
is said to be multiplied by a negative num-' 
ber — b, if it be subtracted b times ; and a 
subtracted b times is — ab. 
4. — a X — b = -\~ a b. Here — a is to 
be subtracted b times; that is, — ab is to be 
subtracted ; but subtracting — a A is the same 
as adding 4” ab; therefore we have to add 
ab. , 
The 2 d and 4 th cases may be thus proved ; 
a — a — o, multiply both sides by b, and a b 
together with — -a X b must be equal to b X 0 , 
or nothing; therefore — a multiplied by b 
must give — ab, a quantity which when ad- 
ded to ab makes the sum nothing. 
Again, a — a — 0 ; multiply both sides by 
— b, then — ab together with — a X — b 
must be — 0 ; therefore — a X — A — 4~ 
a b. 
If the quantities to be multiplied haye co- 
efficients, these must be multiplied together 
as in common arithmetic; the sign and the 
literal product being determined by the pre- 
ceding rules. 
Thus, 3 a X 5 A = 15 a A; because 3 x « 
X5xb=3x5xaxb = \5ab;ixx 
— 11 y — — 44 x y; — 94 x — 5c = -j- 
45 Ac; — 6 d x 4 m = — 24 m d. 
The powers of the same quantity are mul- 
tiplied together by adding the indices ; thus, 
a 2 X a ' —— a ; for aa X aaa — aaaaa. In the 
same manner, a m X a n a m + n ; and — 3 a 2 a: 3 
X 5 a xy 2 — — 15 a 3 x 4 y z . 
If the multiplier or multiplicand consist of 
several terms, each term of the latter must 
be multiplied by every term of the former, 
and the sum of all the products taken, for 
the whole product of the two quantities. 
Ex. 1. Mult, a A -{- x 
by c -j-d 
Ans. ac -f- A c-\-xc-\-ad-\-bd-\-xd 
Here a -j- A -|- x is to be added to itself 
t _J-d times, i. e. c times and d times. 
Ex. 2. Mult, a -\-b — x 
by c — d 
Ans. ac-j-Ac — arc — *d — bd-\-xd 
Here a -|- A is to be taken < — d times ; 
that is, c times wanting d times ; or e times 
positively and d times negatively. 
Ex. 3. Mult, a + A 
. by a 4 - A 
a‘ -|-aA 
a A 4- A 2 
Ans. a 2 -f-2 a A-[- A 2 
Ex. 4. Mult. a?4 ~V 
by x — y 
a 2 4 ~ x y 
xy — y 1 
Ans. x 2 * — y 2 
Ex. 5. Mult. 3a 2 — 5bd 
by — 5 a 2 4 - 4 6 d 
— 15 a 4 -j- 25 a 1 b d 
1 2 a 2 A d — 20 A 2 d 2 
Ans. — 15 a 4 + 35 a 2 A d -— 20 A 2 d 2 
Ex. 6. Mult, a 2 -j- 2 a A A 2 
by a 2 — 1ab-\-h l 
a 4 4" 2 a 3 A -j-u 2 A 2 
— 2 a 3 A — 4 a 2 A 2 — 2 a A 3 
4- a 2i 2 4-2aA 3 4-A« 
Ans. a 4 * — 2 a 2 A 2 * 4" A 4 
Ex. 5. Mult. 1 — ar-j-x 2 — x 3 
by 1 -\-x 
1 — x 4- x 2 — x 3 
— ar 2 4-.T 2 — x 4 
Ans. 1 * * * — x 4 
Ex. 8. Mult, x 2 — p x -j- q 
by ar-|-q 
x 3 — p x 2 -! - ? x 
a x 1 — a p x 4* a q 
Ans. X 3 — p — a.x 2 -\-ij — ap. .r4-a q 
Here the co-efficients of x 2 and x are col- 
lected ; — p — a.x 2 — — p x 2 4- ox 2 ; and 
q — ap.x~qx — ap x . 
r 
, DIVISION. 
To divide one quantity by another, is to de- 
termine how often the latter is contained in the 
former, or what quantity multiplied by the latter 
will produce the former. 
Thus, to divide a A by a is to determine 
how often a must be taken to make up a A ; 
that is, what quantity multiplied by a will 
