16. Find five exact binomial divisors of a6 - x6. Since a6 x6 = (a2)3 — (x2)3, a6 — x6 may be regarded as the difference of two odd powers, and is, therefore, divisible by a2 - x2 (Prin. 1). Since a6 — x6 = (a3)2 – (x3)2, a6 — x6 may be regarded as the difference of two squares, and is, therefore, divisible by a3 – 1⁄23 (Prin. 1). Since a3 - x6 = (a3)2 — (x3)2, a¤ - x6 may be regarded as the difference of two squares, and is, therefore, divisible by a3 + x3 (Prin. 2). Therefore, the exact binomial divisors of a6 26 are a - x, a + x, a2 — x2, a3x3, and a3 + x3. 17. Find an exact binomial divisor of a + xo. SOLUTION = Since a6x6 (a2)3 + (x2)3, a6 + x6 may be regarded as the sum of the cubes of a2 and x2, and is, therefore, divisible by a2 + x2 (Prin. 4). 115. 1. Find the value of x in the equation be — b2 — cx — c2. SOLUTION bx b2 = cx - c2. = 2. Find the value of x in the equation x — a3 = 2 — ax. 15. z6n4n31-3 nz + 2 n-n2. 16. x-3b2-192 b2c3 — 4 cx + 16 c2x = 0. 17. 8 b3 – 18 b2 – 57 b – 2 bx + 7 x + 77 = 0. Solve the following problems: 18. A drover, who had 5 times as many sheep as oxen and as many oxen as horses, sold all for $2300,- the horses at $35 a head, the oxen at $25 a head, and the sheep at $4 a head. What was the number of each? 19. A man paid yearly a certain amount of money for taxes and twice that amount for improvements, and received for rent 3 times as much as he paid out for improvements. If his net gain per year was $300, what were his taxes per year? 20. A owed B a certain sum of money and C twice as much. D owed A 3 times as much as A owed B, and E owed A 5 times the sum A owed B. A found that if he could settle with them all he would have $5000. How much did he owe B and C ? 21. After taking 3 times a number from 11 times the number and adding to the remainder 7 times the number, the result was 12 less than 117. What was the number? 22. A merchant failed in business, owing A 3 times as much as B, C twice as much as A, and D as much as A and B. If the entire debt to A, B, C, and D was $28,000, how much did he owe each? 23. At a certain election there were three candidates for the office of mayor. A received half as many votes as B and 4 times as many as C. If the total vote lacked 25 votes of being 2300, how many votes did each receive? 24. Three boys together had 140 marbles. If the second boy had twice as many as the first and half as many as the third, how many had each ? 25. In a certain school of 600 students there were twice as many Sophomores and 3 times as many Freshmen as Juniors, and 40 more Seniors than Juniors. How many students were there in each class? 26. Divide 25 into three parts such that the first is one third of the second and 5 greater than the third. 27. A, B, and C divided $ 40 so that for every $2 A received, B and C each received $3. What was the share of each? 28. Divide $2200 among A, B, and C, so that B shall have twice as much as A and $200 less than C. 29. Divide $ 351 among three persons so that for every dime the first receives the second shall receive 25 cents and the third a dollar. 30. A man gave equal amounts of money to a school and to a library, and the same amount to a hospital. If to all he gave $28,000, what sum did he give each? 31. When wheat was worth 85 cents a bushel, oats 35 cents a bushel, and corn 60 cents a bushel, a man bought a quantity of wheat, oats, and corn for $67. If he bought twice as many bushels of oats as of wheat, and also three times as many bushels of corn as of wheat, how many bushels of each did he buy? REVIEW 116. Simplify: 1. x2+2a√xy − 3 mn + 4 mn − 4 x3 − 5 a√xy+3x2 +4 a√xу 2. 5x+3x2y+4 xу2 — y3 −√x + 6 y3 + xy2 −√ÿ − 5 x2y — 7 x3 -5xу2+x3+2√x + x2y − 6 y3 + √ÿ−√x − 2 x2y + 3 xy2 + x3. 3. (a-3be + 1 c −76) − ( a + 1 bc + 1c + 3b). 4. (a2x2-4 ay + 4 be+ax) - (b2x2-4 by ax + be). 5. 25 — (213 – 5 x1y + 10 x3y2 – 10 x2y3 + 5 xу1 — y3). - 6. 1a — § x − ( a − x) − (3 b − 4 x − } a) + ¦ a. 16. (m-x) (m + x). 17. (m +a) (m − b). 18. (x − m)(x+ n). 19. (x2+x) (x + 2). 20. (x2+4) (x2 - 3). 26. (a+b) (a" — bTM). 31. (ab) (a + b) (a2 + b2). Square 38. (a3+2a+4a3 +8 a2 + 16 a + 32) (a — 2). 49. (6x-4y) (3 x + 5 y). 50. (3x+ay) (3x+by). Divide: 51. (2 a2x — 5 b2y) (4 a2x — 3 b2y). 52. (6 amn +5p) (6 amn — 3 p). 54. (x+y)(x − y) (x2 + y2) (x2 +y1) (x + y3). 56. (16x+1) (4x2 + 1) (2 x + 1) (2 x − 1). 57. x5 -2x+2x2 + 12 x2 - x - 8 by x+1. 58. 24-42+5x2-4x+1 by 2-3x+1. 59. 28 — 45x+45 x 18 x2+9x-1 by x3-4 x2 + 3 x − 1. 60. a 12 a2a+12 by a3-2 a2+4a-3. 61. 6810b25b+4 by b3 - 2 b2+3b-1. 62. m10-6 m3 +5m-2 by m1 + 2 m3 - 3 m — 2. 63. a-160 a1+127 a3-100 a2-20 a+16 by a3-6a2+5a-4. 64. b1+29b-170 63-61 b2+210b-22 by b'+262-5b-11. 65. 6a2+3% a2y3 — 3 3 ayo + 24y2 by a3 + 1⁄2 a2y − 1 ay2 + } y3. 66. a2c-ab2+acd-ad2-abc+b3 — bed + ba2 — ac2 + cb2 — c2d+cd2 by ac-b2+ cd-d2. |