16. Multiply ax - 3 ay + xy by 3 ay + a x. a x -- 3 ay + xy a' m — 9 ao yö + 3 a yms + a x* y. In adding these two products, the quantity 3 a’ x y occurs twice, with different signs; they therefore destroy each other and do not appear in the result. 17. Multiply 5ad + 3acd-5 ac by 2 ac +2 ad 18. Multiply 13 ary-2 ab ye + 3 c yo by 5 cy+ 7 ab ye + 3. 19. Multiply 11ac + 3 aC-4 a® by 2 ac + ac 20. Multiply a' — 2 acte by a + c. 3a-36 by 2 a + 2b 22. 36 +2c by 36. 3b +20 21. 2 a 6 ab +4 ac-9b-6bc. If 3b + 2 c be multiplied by 2 a only, the product will be too large by 3 b times 3b +2c; hence this quantity must be multiplied by 3 b, and the product subtracted from 6 ab + 4 ac. This result may be proved by multiplying the multiplier by the multiplicand, for the product must be the same in both cases. 23. Multiply 2 ad +36c+2 by 4ab- 2 c. Of Thie 24. Multiply 6 a*b+ 2 ab? by 2 a 6-6° — 1. 19 .5 by 9 - 4. = 5 = 14 171 - 45 - 76 + 20 = 191 - 121= 70. 26. Multiply a-6 by Cd. -6 d a с ас — bc -ad+bd a ac ac-bc-ad+bd. This operation is sufficiently manifest in the figures. In the letters, I first multiply a - b by C, which gives a c-bc; but the multiplier is not so large as c by the quantity d, therefore the product ac-bc is too large by d times a – b; this then must be multiplied by d and the product subtracted. -6 multiplied by d gives ad- bd; and this subtracted from - b c gives a c-bc ad + 6d. Hence it appears that if two terms having the sign — be multiplied together, the product must have the sign t. From the preceding examples and observations, we derive the following general rule for multiplying compound quantities. 1. Multiply all the terms of the multiplicand by each term of the multiplier, observing the same rules for the coefficients and letters as in simple quantities. 2. With respect to the signs observe, 1st, That if both the terms which are multiplied together, have the sign +, the sign of the product must be +. 2d, If one term be affected with t, and the other with the product must have the sign — * 3d, If both terms be affected with the sign , the product must have the sign + Or in more general terms, if both terms have the same sign, whether + or, the product must have the sign t, and if they have different signs, the product must have the sign 27. Multiply 3 a b -- 2 ac + 5 by 7 ab- 2 ac - 1. 21 a 13 - 14 a*b c + 35 ab 4 m 3 n. nx. 31. Product 21 a*b* — 14 abc + 35 ab- 6 abc+ 4 a'c-8ac-3ab-5. 28. Multiply 7m + 5n by 29. a + ay- y by -Y 30. n2 + nxt x2 by a2 + ab +62 by a- ab +6. 2 x? — 3 xy + 4y? by 5x - 6xy - 2y. 33. 3 ac-5 ac + 2 c3 2 aạc4 a* c7ac. 34. 2a - a' x + 2 by 30-x-3. 35. 7 a' b + 2b - 1 by 3a - 26 - 1. 32. 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 and a b their difference. Multiply a + b by ab. a + b - 6 a a + ab -ab62 a' -6. That is, if the sum and the difference of two numbers be multiplied together, the product will be the difference of the second powers of these two numbers. Particular Example. Let a = 12 and b = 1. (a + b) x (a - b) = 19 x 5 =95, af 1% – 144—49 = 95. Multiply a + b by a + b. a+b a tob a + ab ab +62 a + 2ab + b. That is, the product of the sum of two numbers, by itself, 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 ab by a 6. The answer is a*, 2 ab + 6", which is the same as the last, except the sign before 2 a b. Multiply a + 2ab + be by a + b, that is, find the third power of a +6. Ans. ai + 3ab + 3 abi + 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 + b by a-b. Ans. a' — 3 a' b + 3 ab* - 63. 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 quantities 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 quo tient, 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. by 3 b, the quotient is 2 a c. If by 3 ab, the quotient is 2 c. If by 6 a the quotient is bc. For in all these instances the quotient multiplied by the divisor, produces the dividend 6 a b c. 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 a b c. 2. If 6 a b c be divided into 2 a parts, what is one of the parts? Ans. 36c; because 2 a times 3 b c is 6 a b c. |