CASE III. When both the factors are compound quantities. RULE. Multiply each term of the multiplicand by each term of the multiplier; then add all the products together, and the sum will be the product required. * In the first example we multiply a+b, the multiplicand, in to a, the first term of the multiplier, and the product is a2+a b; then we multiply the multiplicand into b, the second term of the multiplier, and the product is ab+b2. The sum of these two products is a2+2ab+b2, as above, and is the square of a+b. In the first example, the like terms of the product, viz. ab and ab together make 2ab; but in the second example, the terms +ab andab, having contrary signs, destroy each other, and the product is a2-62, the difference of the squares of a and b. Hence it appears, that the sum and difference of two quantities, multiplied together, produce the difference of their squares. And by the next following example you may observe, that the square of the difference of two quantities, as a and b, is equal to a2-2ab+b2, the sum of their squares minus twice their pro duct. 9. Multiply x+x2y+xy2+y3 by x-y. 10. Multiply x2+xy+y2 by x2-—xy+y3. Ans. x-y. Ans. x+xy+y*. 11. Multiply 3x2-2xy+5 by x2+2xy-3. 2 Ans. 3x+4x3y-4x4x3y2+16xy—15. 12. Multiply 2a2-3ax+4x2 by 5a2—6ax-2x2. Ans. 10a-27a3x+34a2x2-18x3-8x+. ' DIVISION. DIVISION in Algebra, as well as in Arithmetic, is the converse of multiplication, and is performed by beginning at the left and dividing all the parts of the dividend by the divisor, when it can be done; or by setting them down like a vulgar fraction, the dividend over the divisor, and then reducing the fraction to its lowest terms. In division the rule for the signs is the same as in multiplication, viz. if the signs of the divisor and dividend be alike, that is, both or both, then the sign of the quotient must be +; but if they be unlike, the sign of the quotient must be * * Because the divisor, multiplied by the quotient, must produce the dividend. Therefore, 1. When both the terms are ; the quotient must be +, because in the divisor x + in the quotient produces + in the dividend. 2. When the terms are both -; the quotient is also +, bein the divisor X + in the quotient produces in the cause dividend. 3. When one term is + and the other - ; the quotient must be, because + in the divisor X quotient produces in the dividend; or — in the divisor X - in the quotient in the So that the rule is general; like signs give +, and unlike signs give, in the quotient, CASE I. When the divisor and dividend are both simple quantities. RULE. 1. Place the dividend above a line, and the divisor under it, in the form of a vulgar fraction. 2. Expunge those letters, that are common to the dividend and divisor, and divide the coefficients of all the terms by any number, that will divide them without a remainder, and the result will be the quotient required. It may not be amiss to observe, that when any quantity is divided by itself, the quotient will be unity, or 1; because any thing contains itself once; thus x÷x gives 1, and √2ab divided by 2ab gives 1. NOTE 1. To divide any power by another of the same root; subtract the exponent of the divisor from that of the |