3d, If both terms be affected with the sign, the product mus have the sign + Or in more general terms, If both terms have the same sign, whether + or the product must have the sign +, and if they have different signs, the product must have the sign --27. Multiply 3 a 6 + 2 ac + 5 by 7αο 1. 2 ac 21a* - 14 a be + 35 ab. by X. Product 21 a?l? - 14 a'bc + 35 ab- 6abct 4 a*c*-8ac-3a'b-5. 28. Multiply 7m +5n 4m-3 n. 29. a' tayy by ay. 30. ni + 12 6 to by n 31. a2 + ab +62 by a ab +64. 32. 2 x 3 xy + 4y by 5 *** 6 x y-+2.y%. 33. 3ac-5 ac + 2C* 2 aC-4 a' c-7ac. 34. 2 a' - a' x + 2 by Sa-4-3. 35. 7 ab + 26-1 by 3a’ – 26 - 1. It is generally much easier to trace the effect produced by each of several quantities in forming the result, when the operations are performed upon letters, than when performed upon figures. The following are remarkable instances of this. They ought to be remembered by the learner, as frequent use is made of them in all analytical operations. Let a and b represent any two numbers; a + b will be their sum anda, -b their difference. a' -6. That is, if the sum and the difference of two numbers be multiplóed together, the product will be the difference of the second powers of these two numbers. Particular Example. Let a = 12 and b = 7. (a + b) (a - b) = 19 x 5 = 95, af - 5 = 144-49 = 95. Multiply a +b by a + b. ai + 6 a + ab ab + b a + 2ab +62. That is, the product of the sum of two numbers , by uself, or the second power of the sum of two numbers, is equal to the sum of the second powers of the two numbers, added to twice the product of the two numbers. Multiply a-b by amb. The answer is a? 2 ab +6?, which is the same as the last, except the sign before 2 a b. Multiply a* + 2ab + b* by a + b, that is, find the third power of a +b. Ans. a* + 3a+b + 3a ba + 6*. This is expressed in words thus : the third power of the first, plus three times the second power of the first into the second, plus three times the first into the second power of the second, plus the third power of the second. Multiply a - 2ab + b2 by a-b. Ans. a* — 3 a b + 3 ab—. Which is the same as the last, except the signs before the second and last terms. Instances of the use of the above formulas will frequently occur in this treatise. Division of Algebraic Quantities. XIV. The division of algebraic quantitles will be easily performed, if we bear in mind that it is the reverse of multiplication, and that the divisor and quotient multiplied together must reproduce the dividend. The quotient of a b divided by a is b, for a and b multiplied together produce ab. So ab divided by b gives a for a quotient, for the same reason. If 6 a b c be divided by 2 a, the quotient is 3 b c. by 2 b, the quotient is 3 a c. If by 2 c, the quotient is 3 a b. If by 3 b, the quotient is 2 a c. If by 3 ab, the quotient is 2 c. by 6 a the quotient is bc. For in all these instances the quotient multiplied by the divisor, produces the dividend 6 abc. If Examples. 1. How many times is 2 a contained in 6 abc? Ans. 3 b c times, because 3 b c times 2 a is 6.abc. 2. If 6 a b c be divided into 2 a parts, what is one of the parts? Ans. 35c; because 2 a times 3 b c is 6 a b c. by 3 a. by 23 a с Hence we derive the following RULE. Divide the coefficient of the dividend by the coefficient of the divisor, and strike out the letters of the divisor from the dividend. 3. Divide 16 a b c 4. 4. 12 a b c by 5. 20 a b c 10 b c. 6. 18 a b c d by 6 ad. 7. 6 ab. 8. 17 a d by ad. 9. 4 03 by Observe that 4 ais the same as 4 a a d and a' is the same as a a ; 4 a a a divided by a a gives 4 a for the quotient. It was observed in multiplication, that when the same letter enters into both multiplier and multiplicand, the multiplication is performed by adding the exponents, thus amultiplied by a' is a3+* -= a*. In similar cases, division is performed by subtracting the exponent of the divisor from that of the dividend. a divided by a’ is as — ? 10. Divide 6 a® c by 3 a 6. Ans. 2 abc. 11. 35 72 bd. 12. 16 ac 4 ac. 13. 18 x y by 14. 48 a* xcom by 16 a xo me. 15. 72 a po mi 12 ar 16. 60 p* y 60. aps. 18. 120 a pl by rť. The division of some compound quantities is as easy as that of simple quantities. If a + b + c be multiplied by d the product is d(a+b+c) or ad +6d+cd. Therefore if a d+bd+cd be divided by d, the quotient is a+b+co 66 C6 by 6 g. If ad + bd +cd be divided by a + b + c, the quotient is d. When a compound quantity is to be divided, let the quantity, if possible, be so arranged that the divisor may appear as one of the factors, and then that factor being struck out, the other factor will be the quotient. 19. Divide 12 a' b-9 ac by 3 a. 12 a® b-9ac=3a (4 ab— 3c) Ans. 4ab-3c. Observe that a is a factor of both terms, and also 3. Hence the quantity 12 a b - 9 a c, can be resolved into factors; thus 3(4 a 0-3 a c), or a (12 ab-9c), or 3 a (4 ab-3c). In the last form the divisor 3 a appears as one factor, and the other factor 4 ab 3 c is the quotient. Note. Any simple quantity, which is a factor of all the terms of any compound quantity, is a factor of the whole quantity; and that factor being taken out of all the terms, the terms as they then stand, taken together, will form the other factor. 20. Divide Sa 13 — 16 abc by 2 ab-4 ac. 8 a 13 - 16 Q1c=4 ab? (2 ab- 4 a'c.) Ans. 4 ab. 21. Divide 3abc-15 a b’ d + 9 ao bd by 3ab. 22. Divide 15 abc— 30 + 25 a' c d by 5 a' c. 23. Divide 36 a' b* c— 28 a' b c + 40 a® ca by 9 a + 7 a* b c + 10 a 6* c'. 24. Divide 42 a' - 84 a b c by 1—2 abc. Algebraic Fractions. XV. When the dividend does not contain the same letters as the divisor, or but part of those of the divisor, the division cannot be performed in this way. It can then only be expressed. The usual way of expressing division, as has already been explained, is by writing the divisor under the dividend in the form of a fraction. Thus a divided by b is expressed T |