quantity from which it was subtracted (by the rule of addition): the sum arising will be the remainder.

LXXXIII. MULTIPLICATION.

MULTIPLICATION of algebra is also performed by the following general

RULE.

Multiply the coefficients (if any) together, as in Sect. 4. and to their product join the letters, and prefix the proper sign before them, which, when the signs of the factors are alike, that is, both, or both, the sign of the product is +; but when the signs of the factors are unlike, the sign of the product is.

DIVISION of algebraic quantities is the direct contrary to that of multiplication, and consequently performed by direct contrary operations.

RULES.

1. When the quantities in the dividend have like signs to those in the divisor, and no coefficient in either, cast off all the quantities in the dividend that are like those in the divisor, and set down the other quantities with the sign + for the quotient.

2. When the quantities in the dividend have unlike signs to those in the divisor, then set down the quotient quantities, found as in the last rule, with the sign - before them.

3. If the quantities in the divisor cannot be exactly found in the dividend, then set them both down like a vulgar frac tion, and find all the quantities of the same letters that are in the dividend and divisor, and proceed with the co-efficient as in Case 1, Sect. 38.

4. If the quantity to be divided is compound, range its parts according to the dimensions of some one of its letters, and proceed as in Sect. 5.

5. Different powers or roots of the same quantity are divided by subtracting the exponent of the divisor from that of the dividend, and placing the remainder as an exponent to the quantity given.

Divisor. Dividend.

EXAMPLES.

d)ad+6d( (2) -d)-ad-bd

(3) a)aa+ab(

(4) − a)ab( (5) b)+ab-bd( (6) -bc) abcbcd-bef

(7) 7b) 42— db( (8) 2bx) 8abx-18bxc( (9) 2b)ab—bb ( (10) 20a)10ab-15ac(

(11) a-baaa-зaab+3abb-bbb(

(12) a+b)aa+2ab+bb( (13) a+b)aa-bb

14) 3a-6)6a-96(

15) 3x2-4x+5) 18x-45x3+82x2-6x+40( 16) 4z-5a) 48x376ax2-64a2x+105a3( (17) 3x+4a)81x-256a"(

(18) 2x-3a)16+x-72a2x2-81a4(

(19) 2xy/s)Aryxzz

(20) 20/2cy) 60ab10acxy(

(21) x2) x3( (22) a+x11)â+xll

LXXXV. FRACTIONS.

REDUCTION of algebraic fractions is of the same nature, and requires the same management, as that of numbers. A mixed quantity is reduced to an improper fraction by the rules in Sect. 38, Case 3.

EXAMPLES.

a2-ax

(1) Reduce a-x+~

to an improper fraction.

(2) Reduce a+b+ to an improper fraction.

An improper fraction is reduced to a mixed quantity, by the rule in Sect. 38, Case 4.

Fractions of different denominations are reduced to fractions of equal value, and to have the same denominator, by the rule in Sect. 38, Case 5,

Fractional quantities are reduced into their lowest terms by the rule in Sect. 38, Case 1.

The rules for addition, subtraction, multiplication, and division of algebraic fractions, are the same as for numerical fractions; see Sect. 38, 39, 40, and 41.

d

a-b

(1) Divide by (2) Divide by

acd+d

cd

a

a+b

INVOLUTION is the raising of any given quantity to any proposed power.

1. If the quantity proposed to be involved has no index, that is, if it be not itself a power or a surd, the power thereof will be represented by the same quantity under the given index or exponent.

Thus, the cube or third power of x is expressed by x3.

And the sixth power, a+z, by a+z3, &c.

2. But if the quantity proposed be itself a power or surd, it will be involved by multiplying its exponent by the exponent of the proposed power,

Thus, the fifth power of x is x, the fourth power of

-x

ax+y)' is ax +12, the third power of a—x1⁄2 is a−x. 3. A quantity composed of several factors multiplied together is involved by raising each factor to the power proposed. Thus, the square or second power of ax is a2x2, the cube