REVIEW EXERCISE 141. Give the answers to examples 1 to 40 by referring each to the proper type form: Perform the operations indicated in the following, using type forms wherever possible: 41. (a+b)2 + (a − b)2 + (a - b)(a + b). 42. (x2+x+1)(x − 1) — (x2 − x + 1)(x+1). 43. (a+2b-c) (a — c) — (a2+2ab+c2). 44. (x4-203+x2-1)(x+1). 45. (v+w) (vw2 — v2w + v3 — w3). 46. 16(3a-2b)-5(9 a −7 b)-3(a-4b)-11b. 47. (a6+3 a3 + 9) (a3 — 3). 48. (m2 + mn + n2) (m—n). 49. (p-pq+q2) (p + q). 50. (a2+b2+1 ab- a - b)(a+b+1). 51. (x2+ y2+z2 — xy — xz — yz)(x + y + z). 52. (x + y + z)2 — (x + y − z)2 + (x − y + z)2 − (− x + y + z)*. 53. (a+b)(b+c) − (c + d) (d + a) − (a + c)(b − d). 54. 5[3(a+2b-c) + 4 (a − b −c)]— 19(a − b — c). 55. (x2 – y2)(2 x3 — 4 x2y — 5 xуy2). 56. (a2 - b2) (2 a −3b+5 c) - b(3 a2 — 2 ab +3 b2). 57. 15 x2+24 y2 - (3x+2y) (5 x + 6 y). 58. (a2—b2) (2 b−3 a)+(a+b)(8 b—7 a)a. 59. (x+2)2 -(x − 1)2 — 33 (x − 3). HINT: (1+x- 2x2)(1 + x + 2x2)= [(1 + x) − 2 x2][(1 + x)+2x2]. 69. State the two binomials whose product is (1) p2 — q3; (2) m2 + 4 mn + 4 n2 ; (3) a2 — b2c2; (5) m2 + 2 mn + n2 ; - (4) x2+10x+25; (6) 9 x2-24 xy + 16 y2; (7) 1 - 1 a2. 80. A certain fertilizer contains 1 times as much potash as nitrogen and 4 times as much phosphoric acid as nitrogen. Find the amount of each element in 130 pounds of fertilizer. 81. If 10 is added to a certain number, the sum is three times the original number. Find the number. 82. One number is 32 greater than another. When 3 is added to each number the greater is 5 times the smaller. Find the original numbers. SOLUTION. Let x the smaller number. Also x+3= the smaller number increased by 3, 83. If a certain number is multiplied by 8 and the product is increased by 14, the result exceeds 5 times the original number by 28. What is the number? 84. A boy had twice as much money as his sister; but after each had spent 12 cents he found that he had 3 times as much as his sister. How much had each at first? 85. One number is 5 times another. If 15 is added to each number, the greater will be 3 times the less. Find the original numbers. 86. A rectangle is 3 times as long as it is wide. If both dimensions are increased by 4 inches, it will be twice as long as it is wide. Find its dimensions. 87. A rectangle is 3 inches longer than it is wide. If both. dimensions are increased by 3 inches the area will be increased by 54 square inches. Find the dimensions. 88. A box of candy contained a certain quantity at 35 cents a pound, twice as much at 50 cents a pound, and 3 times as much at 55 cents a pound. If the mixture cost $3, how many pounds of each quality did it contain? SOLUTION. Let x the number of pounds @ 35. Let the student complete the solution. 89. A grocer blended a certain quantity of coffee at 35 cents a pound with twice as much at 32 cents a pound and 4 times as much at 25 cents a pound. If the total value was $ 15.92, find the number of pounds of each in the mixture. stamps, 3 times as many 2¢ stamps, and 10 times as many 1 stamps cost $2.00. How 90. A certain number of 4 many of each were bought? VI. DIVISION 142. Division has been defined as the process of finding one of two factors when their product and the other factor are given. The product is the dividend, the given factor is the divisor, and the factor sought is the quotient. 143. What is the rule for dividing signed numbers? (See § 53). ORAL EXERCISE 144. Divide the following: 1. 8÷(-2); -8÷2; -8÷(-2). 2. 186; 18(-6); 18÷(-6). 3. 36(-9); — 36 ÷ (-6); 36÷(−4). |