Simplify the following by uniting like terms: 39. 12 mn3 - 12 mn2 - 11 mn3 + 11 mn2 — 17 x. 40. √a+b+√a+b2 − 2√ a+b−3√a+b2 + √a + b2. 41. 6 x2y-17 x2y — 19 mn + 12 mn — 6 mn — x2y. ADDITION OF POLYNOMIALS 36. Polynomials are added by uniting their like terms. It is convenient to arrange the expressions so that like terms may be in the same vertical column, and to add each column. Thus, to add 26 ab-8 abc - 15 bc, - 12 ab + 15 abc — 20 c2, - 5 ab+10 bc-6c2, and -7 abc +4 bc+c2, we proceed as follows: 26 ab 8 abc 15 bc 7 abc + 4 bc + c2 bc25 c2 Sum. 37.. Numerical substitution offers a convenient method for checking the sum of an addition. To check the addition of -3a+4b+5c and +2a-2b-c assign any convenient numerical values to a, b, and c, e.g. a=1, b=2, c=1, NOTE. While the check is almost certain to show any error, it is not an absolute test, e.g. the erroneous answer equal 7. -a+6b-4 c would also 38. In various operations with polynomials containing terms with different powers of the same letter, it is convenient to arrange the terms according to ascending or descending powers of that letter. 7 + x + 5 x2 + 7 x3 + 5 x5 is arranged according to ascending powers of x. 5a77 a6b+ 4 a4bc - 8 a2bа2 + 7 ab1d +965 + e7 is arranged according to descending powers of a. EXERCISE 12 Add the following polynomials : 1. 5a-6b-7c, -3a+2b-9c, and 8a-5b+11 c. 2. 8x-5y+7z, 5x+9y-8z, -4x-5y+3z, and −14 x +6y-z. 3. 2a2-9b2-3 c2, -5a2+11b2-9 c2, 4 a2 - 3 b2+5 c2, and -6b2-7c2+8 a2. 4. 5-6x-9z+11 v, 7 r-9x-11 z+8v, 4r-8z, and 8x+14 v. - 5. 26m+10x+14 v +z, − 12 m +15 x − 20 z, — 12 x − 5 m -5%, and 11z-7x. 6. 12 m-14p+13z, -4m+3p+y, 7 m-x-5y, -8 m +2x-y, and 10 m-8p+4y. 7.5xy-2yz-7 xz, 8 xy +3 yz-2 z2, -2xy + 4 xz +5 z2, -xy+yz-4 z2. 8. 3a2-2b2+ c2, -2 a2 + b2-3 c2, a2+3b2-2c2, a2+ 2 b2+4 c2. 9. 13(a+b)-5(b+c) + 7 (c + d), 5 (a + b) +9 (b + c) — 8(c+d), -4(a+b)−5 (b+c)+3(c+d), and 14 (a+b) — (c+d) +6 (b+c). 10. 4√x−√y −3√z, 2√y+√ž, and −4√π-√ÿ−√z. 11. aa1+2 a3, 3a5-4 a1+6 a3, -8a5-7 a*+8 a3, and -3a5-9 a1- 16 a3. 12. 1+x-x2, 1−x+x2, −1+2+22, and -x+x*. 13. 8(x+y)2 -7(x + y) +6, −5 (x + y)2 − 3 (x + y) − 5, and 3(x+y)2+9(x + y) +2. 14. a3+3a2b+3ab2+b3, -2 a3-2 b3, a3-2a2b-b3, and - 2 a2b-b3. 15. 21 pq-17 xy +9 y2, − 21 pq - 9 y2+ xy, and y2-pq+ 17 xy. 16. a-b, b-o, c-d, d-e, and e-a. 17. 7 a3 +4 b3 — c3 + d3, c3 — d3 — e3, b3 — a3 + c3, and c3-b3 — a3. 18.-3.5 a-5.7b+1.8 c, 5.3a-4.3 c-3.6b, 11.2b-2.2 c - 7.4 a. 19. x1-2x2 + x3 − 1, 2 x3 — 2x2+1, 2+x2. 20. -2y-11xy-xy2, 4 y3-3x2+2x2y, 7 x2-6xy2+ x3, 7 x3-4 xy2+4x2y. 21. 2a3-a2-a, 4a3-6a2+a, a3 +8 a2+7 a. 22. 3m3 +5m+8, 10 m3 - 6-4 m2, 2 m3 - 2 m -3. . 23. 4 p3+7p-p+1, 6p2-3p3 +p, 7−2p1+p2, -p3+ 4p-p2. 24. b1— b3 + b2-b+1, -2 b1+2 b3-2 b2-2b+2, 3b13b3+3b2-3b+3. 25. 9a+3+16 a3 + a2, 13 a2+5-4a+8 a3, 11a-15+ 7 a2+6 a3. 26. 5a3-4a2b+3ab2-2b3, -4a3+3a2b-2 ab2+b3, 3 a3 -2ab+ab2, 4a+3a2b-2ab+b3. 27. 7a-4b+3a3b-2 a2b2+7 ab3, -7 ab3+4a3b-7 a*+ 3a2b2, b-3a2b2. 28. .6 a1-7b, 3 ab+3a2b2, 6 a2b2-5 a, b1- 6 a3b. SUBTRACTION EXERCISE 13 1. What is the remainder if 6 is taken from 12? 2. If from the 6 negative unfts, -1, -1, -1, −1, −1, -1, four negative units are taken, how many negative units remain? What is therefore the remainder when - 4 is taken from 6? 3. Instead of subtracting in the preceding example, what number may be added to obtain the same result? 4. The sum total of the units +1, +1, +1, +1, +1, −1, -1, and 1, is 2. What is the value of the sum if two negative units are taken away? If three negative units are taken away? 2? 5. What is therefore the remainder when 2 is taken from When 3 is taken from 2? 6. What other operations produce the same result as the subtraction of a negative number? 7. If you diminish a person's debts, does he thereby become richer or poorer? 8. State other practical examples which show that the subtraction of a negative number is equal to the addition of a positive number. 39. Subtraction is the inverse of addition. In addition, two numbers are given, and their algebraic sum is required. In subtraction, the algebraic sum and one of the two numbers is given, the other number is required. The algebraic sum is called the minuend, the given number the subtrahend, and the required number the difference. Therefore any example in subtraction may be stated in a different form; e.g. from 5 take 3, may be stated: What number added to -3 will give -5? To subtract from a the number b means to find the number which added to b gives a. Or in symbols, if a−b=x, x + b = a. 40. The results of the preceding examples could be obtained by the following Principle. To subtract, change the sign of the subtrahend and add. The numerical results of Exs. 1-3 of course do not prove this principle, but it may be deduced as follows: The principle is obviously correct for a positive subtrahend. To find a -(—b) we have to find the number which added to b will give the result a. But a+b added to b, gives a. Hence the required remainder is a+b, or, NOTE. The student should perform mentally the operation of changing the sign of the subtrahend; thus to subtract - 8 a2b from – 6 ab, change mentally the sign of -8a2b and find the sum of 6 a2b and + 8 a2b. |