Multi Plicatiox

MULTI PLICATIOX.

The multiplication of simple algebrai cal quantities must be represented ac cording to the notation already pointed out.

Thus, a)( b, or a 8, represents the pro duct a multiplied by b ; a b c, the pro duct of the three quantities, a, b, and c.

It is also indifferent in what order they are placed, a X b and b X a being equal.

To determine the sign of the product, observe the following rule.

If the multiplier and multiplicand haw the same sign, the product is positive; if they have different signs, it is negative.

1. -.1:a x + b r=ab; because in this case a is to be taken positively b times ; therefore the product a I, must be posi tive.

2. —a x-1- b — a b; because — a is to be taken b times ; that is, we must take 3. --Fax— b = — ab; for a quantity is said to be multiplied by a negative num ber — b, if it be subtracted 6 times ; and a subtracted b times is — ab.

4. — a X — b=+ab. Here — a is to be subtracted b times; that is, — a 6 is to be subtracted ; but subtracting — a b is the same as adding + ab ; therefore we have to add + ab.

The and cases may be thus prov ed: a — a=0, multiply both sides by 6, and a to.„cether with — a must be equal to or nothing; therefore, — a mul tiplied by it must giN e a cianti•y which when added to at, 'makes the sum nothing.

Again, a — a=o ; multiply both sides by — 6, then — a b together with — X —6 must be =o ; therefore —a X — b =+ab.

If the quantities to be multiplied have co-efficicnts, these must he multiplied to gether, as in Common arithmetic; the sign and the literal product being determined by the preceding rules.

Thus, 3a x 5 b== 15ab ; because 3 X a X 5X6=-3X5XaXb=15ab; 4x x —11y — 44r.y ; — 9bx— 5c= +45bc ; — 6d ,(4nt= — 24 in d.

The powers' of the same quantity are multiplied together by adding. the indices ; thus a' x a3 =a5 ; for aa aaa = aaaaa. In the same manner, an' X an =arn+n and — 3a. x3 y5a x y'= — 15a3 x4 y5.

If the multiplier or multiplicand con sist of several terms, each term of the lat ter must be multiplied by every term of the former, and the sum of all the pro ducts taken, for the whole product of the two quantities.

Ex. 1. Mult. a+b+x by c+d Ans. a c+b c+x c+a d+ b el+x d Here a + b + x is to be added to itself c+d times, i. e. c times and d times.

Ex. 2. Malt. a + b — x by c — d Ans. a c+b c—x c—a d—b d+x Here a+b is to be taken c — d times, that is, c times wanting dtimcs ; or c times positively and d times negatively.

Ex. 3. Mult. a+b by a+b .a'+a b +a b+b' Ans. a3+2 a b-4-b.

Ex. 4. Mult. x+y by x'±x y —x y — y.

Ans. x' — a' Ex. 5. Mult. 3 a,— 5 b d by — 5 al+ 4 b d —15 a4+25 a' b d +12 a. b d — 20 b' Ans. — 15 a++:37 a' b d — 20 b' d' Ex. 6. Mutt a.+2 a b+101 by a.-2 a 6+6' a4+2a3b +a= bl —2a3b —41.b.-2ab3 + o'b'+ 2ab3+b4 Ans. as • — 2a.b. * +b+ Ex. 7. 3Iult. 1 — x+x. x.; by 1 + x 1 — x+x. — x3 x—x, x3 — x+.

Ans. 1 * * * — x.+4 —Ex. 8. Mult. x. — p x q by x + a x3 x +ax'—apx+aty Ans. x3 — ap.x Here.the co-efficients of x. and x are collected ; — p — a . x. x" a x'; and q —ap.;x=2.x—aps.

Divistoy.

To divide one quantity by another, is to de-, termine how often the latter is contained in the former, or what quantity tmdtiplied by the latter will produce theformer.

Thils,to divide a b by a is to determine how often a must be taken to make up a It ; that is, what quantity multiplied by a will give a b ; which we 'know is b. From this consideration are derived an the rules for the division of alge,braical quantities.

If the divisor and dividend be affected with like signs, the sign of the quotient 'is + : but if their signs be uniike, the sign of the quotient is —.

If — a b be divided by — a, the quo tient is + b ; because — a X + b gives — a b ; and a similar proof may be given in the other cases.

In the division of simple quantities, if the co-efficient and literal product of the divisor be found in the dividend, the other part of the dividend, with the sign deter mined by- the last rule, is the quotient.