By Arthur Wouk
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Additional info for A course of applied functional analysis
What is the degree of (a)? In Problems 9–14, is the algebraic expression a polynomial? If so, give its degree. 9. 4 Ϫ x2 10. x3 Ϫ 5x6 ϩ 1 2. What is the degree of (b)? 11. x3 Ϫ 7x ϩ 81x 12. x4 ϩ 3x Ϫ 15 3. What is the degree of the sum of (a) and (b)? 13. x5 Ϫ 4x2 ϩ 6Ϫ2 14. 3x4 Ϫ 2xϪ1 Ϫ 10 4. What is the degree of the product of (a) and (b)? 5. Multiply (a) and (b). 6. Add (a) and (b). 7. Subtract (b) from (a). 8. Subtract (a) from (b). In Problems 15–22, perform the indicated operations and simplify.
5 ϫ 10Ϫ6 23. 99 ϫ 105 24. 75 ϫ 1011 25. 1 ϫ 10Ϫ7 26. 167 ϫ 10Ϫ4 12y 16y 5 6 7 11 81. x 2 3xy 84. 3 1y 2 1y Ϫ 3 79. 1 77. 3 1 7 4 16 Ϫ 2 3 82. 2a 2 8a8b13 85. 2 15 ϩ 312 5 15 ϩ 212 3 3 254 80. 12 16 ϩ 2 83. 12m15 120m 86. 312 Ϫ 213 313 Ϫ 212 2 In Problems 27–32, change to radical form. Do not simplify. 27. 321ր5 28. 6253ր4 29. 4xϪ1ր2 30. 32yϪ2ր5 31. x1ր3 Ϫ y1ր3 32. (x Ϫ y)1ր3 87. What is the result of entering 23 on a calculator? 2 88. Refer to Problem 87. What is the difference between 2(3 ) and 3 2 32 (2 ) ?
Indicate true (T) or false (F), and for each false statement find real number replacements for a, b, and c that will provide a counterexample. For all real numbers a, b, and c: (A) (a ϩ b) ϩ c ϭ a ϩ (b ϩ c) (B) (a Ϫ b) Ϫ c ϭ a Ϫ (b Ϫ c) (C) a(bc) ϭ (ab)c (D) (a Ϭ b) Ϭ c ϭ a Ϭ (b Ϭ c) In Problems 41–48, indicate true (T) or false (F), and for each false statement give a specific counterexample. 25. 3(xy ϩ z) ϩ 0 ϭ 3(xy ϩ z) 26. ab(c ϩ d ) ϭ abc ϩ abd Ϫx x 27. ϭ Ϫy y 39. Indicate true (T) or false (F), and for each false statement find real number replacements for a and b that will provide a counterexample.