32. When the dividend is not a multiple of the divisor. Set them down as a vulgar fraction, the divisor being the denominator; and divide the terms by such numbers and quantities as are common to both; the result will be the fractional answer. When the dividend is a compound quantity, and the divisor a simple one. Divide every term of the dividend by the divisor, as in the former case, and connect the results by their proper signs. EXAMPLES. 18ax-15a y+27a3 3a =6x-5ay+9a2 1. Divide 15axy3-30bx3y+10x2y by 5xy. CASE 3. 35. When the divisor and dividend are compound quantities. Arrange the terms, both of the divisor and dividend, in such manner, that the higher powers of some one letter shall always precede the lower; then find the first term of the quotient, by using the first of the divisor and dividend, as in the first case. Multiply the whole divisor by the quotient thus found; subtract the product from the dividend, as in common arithmetic; and proceed till the dividend is exhausted. The remainder, if any thing remain, with the divisor for a denominator, must be annexed, with its proper sign, to the quotient. EXAMPLES. a+b)a3+3a2b+3ab2+b3 (a2+2ab+b2 2a2b+3ab2 ab+b3 a2-2ab+b2)a5-5ab+10 a3b2-10 a2b3+5ab3b3(α3 a5-2ab+ a3b3 [Sab+3ab2b3— 265 -3a+b+ 9a3b2-10a2b3 a2-2ab+b2 —3a+b+ бa3b3- 3a2b3 Divide x-4x3y+6x3y2—4xy3+y1 by x—y. Divide a2-b2 by a—b. Divide a3+b3 by a+b. Result, a+b. Result, a2ab+b2. Result, y-3. Divide y3-3y3-4y+12 by y2-4. Divide b-3b+x2+b2x2-x by b3-3b3x+3bx2—x3 Result, b3+362x+3bx2+x3. Result, x-x+y+x3y®—x2y3+xya—y3+ x+y PROMISCUOUS EXAMPLES. 1. Required the sum of 5ac+7be, Sac-4be, 7ac-3be, 8ac+4be, and 2ac-3be. Ans. 25ac+be. 2. Collect Sax+5xy-72°, Sxy-2ax+4x2, 7ax+3z2 -xy, and 52-4ax+2xy, into one sum; and subtract therefrom the sum of the following quantities, Sax-5xy +4z2,7xy-5αx-622, and 5ax-2xy+622. Result, ax+9xy+z3. 3. Required the product of x3+x2y+xy2+y3 by x—y. Ans.xy1. 4. What is the sum of the products of x+Sxy+3xy +ys by x+y, and x-2xy+y2 by x2-2xy+y2?. Ans. 2x+12x3y3+2y*. 5. Divide 25-ys by x-y, and from the quotient subtract the product of x+y by x—xy. Result, 2x3y+2xy3+y*. 6. Multiply 3a-9ab9ab2-368 by a2+2ab+b2, and a3+3a2b+3ab2+b3 by 3a2—6ab+3b3, and find the sum Result, 6a-12a3b2+6ab3. of the products. 7. If a1-b1, be divided by a-b, and the product of a2+ab+b2 by a-b, subtracted from the quotient, what will the remainder be? Ans. ab+ab+2b3. 8. Divide x+8x7y+28x®y2+56x3y3+70x1y1+56x3ys 28x2y®+8xy2+y3 by x2+2xy+y2; and multiply *—4x3y +6x2y2—4xy3+ya by x2—2xy+y3, and find the sum and difference of the results. Result, {Sum, 2x+50x+y3+30x2y*+2¥3. Difference, 12x5y+40x3y3+12xy.. SECTION II. INVOLUTION, OR THE RAISING OF POWERS. 34. To involve a given quantity to any power.* Multiply the quantity by itself as many times as there are units in the index of the given power diminished by one.. EXAMPLES. Required the 5th power of a3bc. (a3b°c)=a15b10c5 Required the 4th power of 2b-ç. *When the quantity to be involved is a simple one, multiply its index, or indices by that of the power proposed, observing that the even powers of negative quantities are positive, and the odd ones negative. Required the 6th power of 5a+b. Required the 3d power of a+2b+c. 35. A binomial is raised to any power, with great facility, by the following method. Set down, for the first term of the power, the first term of the root involved to the given power. The succeeding terms, without the co-efficients, consist of the successive powers of the first term of the root, regularly descending, joined to the powers of the second, regularly ascending from the first; the common difference of the indices being one. The co-efficient of the second term is the index of the given power; and if a co-efficient, already found, be multiplied by the exponent of the leading quantity, or first |