The utility of these theorems will be the more apparent the more complicated the expressions to which they are applied. Frequent examples of their application will be seen hereafter. (68.) The same theorems will enable us to resolve many complicated expressions into their factors. Ex. 1. Resolve a’ +2ab+b? into its factors. Ans. (a+b)(a+b). Ex. 2. Resolve n°+2n+1 into its factors. Ans. (n+1)(n+1). Ex. 3. Resolve a’-2ab+6° into its factors. Ans. (a−b)(a-6). Ex. 4. Resolve a’-6ab+962 into its factors. Ans. (a–36)(a-36). Ans. la 27 Ex. 5. Resolve a'-6° into its factors. Ans. (a+b)(a–6). Ex. 6. Resolve a*-64 into its factors. Ans. (a’ +)(a?—b). QUEST.—How may complicated expressions be resolved into factors ? SECTION V. DIVISION. (69.) Division consists in finding how many times one quantity is contained in another. The quantity to be divided is called the dividend ; and the quotient shows how many times the divisor is contained in the dividend. When we have obtained the quotient, we may verify the result by multiplying the divisor by the quotientthe product should be equal to the dividend. Hence we may regard the dividend as the product of two factors, viz., the divisor and quotient-of which one is given, that is, the divisor; and it is required to find the other factor, which we call the quotient. CASE I. (70.) When the divisor and dividend are both moolomials. Suppose we have 72 to be divided by 8. We must find such a factor as multiplied by 8 will give exactly 72. We perceive that 9 is such a number, and therefore 9 is the quotient obtained when we divide 72 by 8. Also, if we have ab to be divided by a, it is evident that the quotient will be b; for a multiplied by b gives the dividend ab. Quest.-What is the object of division? What is the dividend ? What is the divisor? What is the quotient? What is case first? If we divide 60x by 5, we obtain 12x, for 12x multiplied by 5 gives 60x. So, also, 12mn divided by 3m gives 4n. Again, suppose we have a' to be divided by a'. We must find a number which, multiplied by a', will produce a'. We perceive that a® is such a number; for, according to Art. 56, we multiply a' by a' by adding the exponents, 2 and 3 making 5. That is, the exponent 3 of the quotient is found by subtracting 2, the exponent of the divisor, from 5, the exponent of the dividend. Hence we derive the following RULE OF EXPONENTS IN Division. (71.) A power is divided by another power of the same root, by subtracting the exponent of the divisor from that of the dividend. Examples (1.) Divide (6.) (7.) (8.) Divide by ho Quotient hi (72.) Let it be required to divide 48a by 6a. We must find a quantity which, multiplied by ba', will produce 48a'. Such a quantity is 8a®; for, according to Arts. 55 and 56, 8a' x 6a’ is equal to 48a'. There QUEST.-How are powers divided ? |