10aa—27a3x+34a2x2—18ax3—8xa 10x3—7ax1—a2x3—3a3x2 * +a3 25—(a+d)x2+(b+ad+e)x3—(c+bd+ae)x2+(cd+be)x—ce Ex. 16. 114-1962-(x2+24)2}or, 114-(196x2x2-18x2-576) or mult. 114-(148x2-x576) by 114-(148x2—x1—576) 1142-114(148x2-x1—576) -114(148-x-576)+(148x-x-576) Prod. 12996-228/(148-x-576)+(148x-x-576) NOTE. When the quantities that are to be multiplied together efficients, pro have literal co ax+b cx-d ceed as before, acx2+bcx -adx-bd sum, or differ ence of the co acx2+(bc-ad)x-bd a-bx c-dx ac -bcx -adx+bdx2 ac-(bcad)x+bdx2 efficients of the resulting terms, between brackets, as in the former rule. And if several compound quantities are to be multiplied together, multiply the first by the second, and then that product by the third, and so on to the last factor, as below. To this we may add, that it is usual, in some cases, to write down the quantities that are to be multiplied together, between brackets, or under a vinculum, without performing the whole operation; as 3ab(a+b)×an (a2—b2) 110638080-9408001883600-8000x8000.x -55319040x+4704002-941800x4000x-4000x 1111084802-55319040x-1882600+1887600x*-12000+80002 Ex. 20. 2352+20x2-10x multiply 5531904+47040-23520x a3-3a2x+3ax2-x3 Ex. 22. a3+3a2x+3ax2+x3 a°+3a3x+3a2x2+a3x3 -3ax-9a1x2-9a3x3-3a2x2 +3a2x2+9a3x3+9a2x2+3ax3 a-3a2x* *+3a2x2 * Ex. 23. 72-1676x840x-21952 multiply -11732x28089762-1407840x+36791552x 5880-1407840270560022-18439680.x -153664x+367915522-18439680x+481890304 49x6-23464x+2820736x-3123008x+742887042-36879360x+481890304 NOTE. The products of the powers of the same quantity are found by adding their indices. Thus: a'Xa'a2, or a Xu= m+n a5; am Xa"-am 3+4 12 1+1 I I 2 a1Xa-al- ; ¿1+2 — a3 ; a2 Xa3= m+n ; a" Xa"am "; xxx=x+1. a Xa*a* ; Obs. 1. In multiplication, as well as in addition and subtraction, the order of the letters is of no consequence. Thus, if abc be multiplied by d, the product is abcd, bacd, cbda, &c., each of which is of the same value; but it is usual to arrange them according to the order of the alphabet. Obs. 2. In algebra it is customary to begin the multiplication on the left; but because the steps are merely indicated, it is of no consequence where the opera tion commences. Obs. 3. It may be useful to observe, that, according to Euclid, Lib. II. Prop. V., the product of the sum and difference of any two quantities is equal to the difference of their squares; thus, (1.) (a+b) (a-b)—a2-b2 (a+b)(ab). (2.) (a2+b2) (a2—b3)—a*b*—(a2+b2) (a+b) (a-b). (3.)(a+b)(a-b1)—a— c3 — b2 = (a + + b 1) (a2 + b2) (a2 —b3)=(a*-+-b1) (a2+b2) (a+b) (a—b) These compositions and decompositions of quantities are often found to be of great utility in the solution of equations. Ex. 24. Multiply ex+mx2 + nx2 + rx2 bex + bmx+bnx+brx® dex+dmx+dnx+drx3 aex2+(am+be)x2+(an+bm+ce)x2+(ar+bn+cm+de)x +(br+en+dm)x+(cr+dn)x2+drx3 DIVISION. 41. Division in Algebra is the method of finding the quotient arising from the division of one algebraic quantity by another. Division is generally divided into three cases, namely, when the divisor and dividend are both simple quantities; when the divisor is a simple quantity and the dividend a compound one; and when the divisor and dividend are both compound quantities. CASE I. When the divisor and dividend are both simple terms. RULE. Place the divisor in the form of a denominator under the dividend; cancel those letters which are common to both, and divide the coefficients by any number that will divide them without a remainder, and the result will be the quotient required. RULE II. Divide the coefficients as in common arithmetic, and to the quotient annex those letters in the dividend which are not found in the divisor. A general rule for the signs in all the cases of division: When the signs of the divisor and dividend are alike, (that is, both or both,) the sign of the quotient will be +. When + they are unlike, (that is, the one and the other ,) the sign of the quotient will be -. The above rule briefly expressed in one view, is as follows: Div'r. Div'd.、 Quo't. } + { } Both minus. Div'r. Div'd. Quo't. + 1. Plus } Plus { + Plus 3 +ab = +b; = +b; b - b; +a +a a and these four are all the cases that can possibly happen with regard to the variation of the signs. Powers and roots of the same quantity are divided by subtracting their idices, that is, subtract the index of the divisor from the index of the dividend. |