CASE I. When the factors are both simple quantities. RULE. Multiply the coefficients of the two terms together, and to the product annex all the letters, or their powers, belonging to each, after the manner of a word; and the ; result, with the proper sign prefixed, will be the product required (d). (d) When any number of quantities are to be multiplied together, it is the saine thing in whatever order they are placed: thus, if ab is to be multiplied by c, the product is either abe, neb, or bca, CASE II. When one of the factors is a compound quantity. RULE. Multiply every term of the compound factor, considered as a multiplicand, separately, by the multiplier, as in the forntres case; then these products, placed one after another, with their proper sighs, wir be the whole product required. &c. ; though it is usual, in this case, as well as in addition and subtraction, to put them according to their rank in the alphabet. It Isay here also be observed, in conformity to the rule given ab sve for the signs, that (+a)x(+0), or (-a)X(-6)= to ab; and (ta) x (-6), or (-a)x(+6) ab. CASE III. When both the factors are compound quantities. RULE. • Multiply every term of the multiplicand separately, by each term of the multiplier, setting down the products one after another, with their proper signs; then add the several lines of products together, and theirum will be the whole product required. 1. Required the product of x-xy+yz and xty. 2. Required the product of 234xytoya +43 and -Y: C% ! 3 Required the product of x2 + xy+ya and x? xy+ya. 4. Required the product of 3x2 – 2xy +5 and x3 + 2xy - 3. 5. Required the product of 2a2 - 3ax+4x2 and 5a2 .- 6ах - 2x2. 6. Required the product of 5x3+4ax2 +3a2stas, and 2x3 - 3ax taa. 7. Required the product of 3x3+2xoys +3y3 and 2x3 3x3y2 + 5y3. &. Required the product of 33 - ax+-bx - and 2-dx+e. DIVISION Division is the converse of multiplication, and is performed, like that of numbers; the rule being usually divided into three cases ; in each of which like signs give + in the quotient, and unlike signs -, as in findin t ) their pčoducts (e). It it libre also to be observed, that powers and roots of the same quantity, are divided by subtracting the index of the divisor from that of the dividend. Thus, de raa, or =a; at_a}, or, = ad'; , = 2 tab - ab fr) According to the rule here given for the signs, it follows that ab tab =ta, = So, Q, -- 6 as will readily appear by multiplying the quotient by the divisor; the signs of the products being then the same as would take place in the former rule. CASE I. When the divisor and dividend are both simple quantities. RULE. Set the dividend over the divisor, in the manner of a fraction, and reduce it to its simplest form, by cancelling the letters and figures that are common to each term. a a, or - a 1 a - 2a Also – 2a; 3a, or = - f ; and 9c13x1=3z* За , 5 CASE II. When the divisor is a simple quantity, and the dividend a compound one. RULE. ivide each term of the dividend by the divisor, as in the former case ; setting down such as will not divide in the simplest form they will admit of. |