#18 from textbook b My Questions | bartlebyThomas W. Hungerford - Abstrac ×+O File | C:/Users/angel/Downloads/Thomas%20W.%20Hungerford%20-%20Abstract%20Algebra_%20AN%20lntroduction-Cengage%20Learning%20(2014).pdf-- A' Read aloud V DrawF HighlightO Erase133of 621(b) If f(x), g(x) e F[x] have the same roots in F, are they associates in F[x]?14. (a) Suppose r, s E Fare roots of ax? + bx +c€ F[x] (with a + 0p). Use theFactor Theorem to show that r +s = -a-b and rs = a'c.Capt 2012 C La ARig Ra d May act be pind caad or dy t in wale or in part Dee te eied,d ttg d dday ate o ingpe Cleng driBodk dr . Rartal v badatt4.4 Polynomial Functions, Roots, and Reducibility111(b) Suppose r, s, t e Fare roots of ax + bx² + cx + de F[x] (with a + Op).Show that r + s+ t= -ab and rs + st + rt = acand rst = -a'd.15. Prove that + 1 is reducible in Z[x] if and only if there exist integers a and bsuch that p = a + b and ab = 1 (mod p).16. Let f(x), g(x) e F[x] have degree sn and let co, C1, . .., C, be distinct elementsof F. If f(c) = g(c) for i = 0, 1, ..., n, prove that f(x) = g(x) in F[x].17. Find a polynomial of degree 2 in Zx] that has four roots in Zg. Does thiscontradict Corollary 4.17?18. Let p:C → C be an isomorphism of rings such that o(a) = a for eacha E Q. Suppose re Cisa root of f(x) E Q[x]. Prove that o(r) is also aroot of f(x).19. We say thata E Fis a multiple root of f(x) E F[x] if (x – a)k is a factor off(x) for some k 2 2.(a) Prove that a e R is a multiple root of f(x) E R[x] if and only if a is aroot of both f(x) and f"(x), where f"(x) is the derivative of f(x).(b) If f(x) E R[x] and if f(x) is relatively prime to f"(x), prove that f(x) hasno multiple root in R.20. Let R be an integral domain. Then the Division Algorithm holds in R[x]whenever the divisor is monic, by Exercise 14 in Section 4.1. Use this fact toshow that the Remainder and Factor Theorems hold in R[x].21. If R is an integral domain and f(x) is a nonzero polynomial of degree n inR[x], prove that f(x) has at most n roots in R. [Hint: Exercise 20.]22. Show that Corollary 4.20 holds if Fis an infinite integral domain. [Hint: SeeExercise 21.]23. Let f(x), g(x), h(x) e F[x] and re F.(a) If f(x) = g(x) + h(x) in F[x], show that f(r) = g(r) + h(r) in F.(b) If f(x) = g(x)h(x) in F[x], show that f(r) = g(r)h(r) in F.Where were these facts used in this section?2:58 PMO Search for anythingEPICAiEPIC11/20/2020

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Given: Let ϕ:ℂ→ℂ be an isomorphism or rings such that ϕ(a)=a for each a∈ℚ. Suppose r∈ℂ is the root…

Answered: 18. Let p:C + C be an isomorphism of…

Math

Advanced Math

18. Let p:C + C be an isomorphism of rings such that e(a) - a for each ae Q. Suppose r E Cisa root of fx) E Q[x]. Prove that p(r) is also a root of f(x).

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.8: Probability

Problem 32E

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Consider a screening system that analyzes the luggage that is being loadedto a plane. Each package first goes through a screening device. If it passes the test there, then it is loaded to the plane. If it does not, the package goes to a second screening device. The procedure is similar for the second device: if the luggage passes the test, it is loaded to the plane, if it fails, it issent to a third device. Packages that passes the test at third device are loladed to the plane, the ones that fail are taken paprt and not loaded to the plane.The screening devices i = 1; 2; 3 are not perfect. For each device, there are two risks:– With probability pi, each device erroneously passes a bag that is dangerous, given that previous devices have correctly rejected the bag.– With probability qi, each device erroneously fails a harmless bag that previous devices have also erroneously failed.Assume that pi; qi > 0 What is the probability that a dangerous package will be loaded to the aircraft?…
High school seniors with strong academic records apply to the nation’s mostselective colleges in greater numbers each year. Because the number of slotsremains relatively stable, some colleges reject more early applicants. Suppose thatfor a recent admissions class, an Ivy League college received 2851 applications forearly admission. Of this group, it admitted 1033 students early, rejected 854outright, and deferred 964 to the regular admission pool for further consideration.In the past, this school has admitted 18% of the deferred early admission applicantsduring the regular admission process. Counting the students admitted early andthe students admitted during the regular admission process, the total class sizewas 2375. Let ?, ?, and ? represent the events that a student who applies for earlyadmission is admitted early, rejected outright, or deferred to the regularadmissions pool.a. Are events E and D mutually exclusive? Find (? ∩ ?)b. Suppose a student applies for early admission. What…
High school seniors with strong academic records apply to the nation's most selective colleges in greater numbers each year. Because the number of slots remains relatively stable, some colleges reject more early applicants. Suppose that for a recent admissions class, an Ivy League college received 2851applications for early admission. Of this group, it admitted 1033 students early, rejected 854 outright, and deferred 964 to the regular admission pool for further consideration. In the past, this school has admitted 18% of the deferred early admission applicants during the regular admission process. Counting the students admitted early and the students admitted during the regular admission process, the total class size was 2375. Let E, R, and D represent the events that a student who applies for early admission is admitted early, rejected outright, or deferred to the regular admissions pool. a. Use the data to estimate P( E), P(R), and P( D). b. Are events E and D mutually exclusive?…
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It is known from experience that in a certain indus-try 60 percent of all labor–management disputes are over wages, 15 percent are over working conditions, and 25percent are over fringe issues. Also, 45 percent of the disputes over wages are resolved without strikes, 70 per-cent of the disputes over working conditions are resolved without strikes, and 40 percent of the disputes over fringe issues are resolved without strikes. What is the probabil-ity that a labor–management dispute in this industry will be resolved without a strike?
Suppose Martin is a very talented used-car salesman. Whenever Martin talks to a new customer, there is a 90% chance that he convinces the customer to purchase one of his used cars. Brian, Martin's boss, is envious that Martin sells many more cars than he does. Because of his jealousy, Brian institutes a new rule that Martin is only allowed to talk to 35 customers per day. Thus, Martin continues to work each day until he speaks to 35 customers, at which point Brian sends him home. Let ? represent the number of used cars that Martin sells on a given day. What are the mean, ?, and variance, ?^2, of ?? Please round your answers to the nearest two decimal places. ?=_______? ?^2=_________?
Consider a town with a population that is 10 percent black (and the remainder is white). Because blacks are more likely to work the night shifts, 20 percent of all cars driven in that town at night are driven by blacks. One out of every 20 people driving at night is drunk, regardless of race. Persons who are not drunk never swerve their car, but 10 percent of all drunk drivers, regardless of race, swerve their cars. On a typical night, 5,000 cars are observed by the police force. a. What percent of blacks driving at night are driving drunk? What percent of whites driving at night are driving drunk?b. Of the 5,000 cars observed, how many are driven by blacks? How many of these cars are driven by a drunk? Of the 5,000 cars observed at night, how many are driven by whites? How many of these cars are driven by a drunk? What percent of nighttime drunk drivers are black?c. The police chief believes the drunk-driving problem is mainly due to black drunk drivers. He orders his policemen to…
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