#9 from book Thomas W. Hungerford - Abstrac xb My Questions | bartlebyO File | C:/Users/angel/Downloads/Thomas%20W.%20Hungerford%20-%20Abstract%20Algebra_%20AN%20lntroduction-Cengage%20Learning%20(201.....Flash Player will no longer be supported after December 2020.Turn offLearn moreof 621+-- A Read aloudV DrawF HighlightO Erase122(g) 2x* + 5x³ – 5x - 2 and 2x – 3x² – 2x in Q[x].6. Express each of the gcd's in Exercise 5 as a linear combination of the twopolynomials.B. 7. Let f(x) E F[x] and assume that f(x)|g(x) for every nonconstant g(x)E F[x]. Showthat f(x) is a constant polynomial. [Hint: f(x) must divide both x + 1 and x.]8. Let f(x), g(x)E F[x], not both zero, and let d(x) be their gcd. If h(x) is acommon divisor of f(x) and g(x) of highest possible degree, then prove thath(x) = cd(x) for some nonzero cEF.%3D9. If f(x) # OF and f(x) is relatively prime to Or, what can be said about f(x)?10. Find the gcd of x + a + b and x3 – 3abx + a³ + b³ in Q[x].11. Fill in the details of the proof of Theorem 4.8.12. Prove Corollary 4.9.13. Prove Theorem 4.10.Coutseome1.14. Let f(x), g(x), h(x)E F[x], with f(x) and g(x) relatively prime. If f(x)|h(x) andg(x) |h(x), prove that f(x)g(x)|h(x).9:16 AMO Search for anythingEPICAiEPICO 4)3412/7/2020

Name: #9 from book Thomas W. Hungerford - Abstrac xb My Questions | bartleby
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Description: #9 from book Thomas W. Hungerford - Abstrac xb My Questions | bartlebyO File | C:/Users/angel/Downloads/Thomas%20W.%20Hungerford%20-%20Abstract%20Algebra_%20AN%20lntroduction-Cengage%20Learning%20(201.....Flash Player will no longer be supported afte

Relatively prime elements: Two elements f(x), g(x) ∈ F[x] are said to be relatively prime if GCD(f(x), g(x)) = 1. Given: f(x) and 0F are realtively prime.Since, f(x) and 0F are realtively prime. ⇒GCD(f(x), 0F) = 1. We know that GCD is greatest common divisor. Since, GCD(f(x), 0F) = 1 this means greatest common divisor of f(x) and 0F is 1. Also, we know that every polynomial in F[x] is divisor of 0F. Because, 0F = 0F · h(x) for all h(x) ∈ F[x]. Therefore, set of common divisors of f(x) and 0F is set of divisors of f(x). We know that f(x) divides f(x), and if h(x) is any other polynomial which divides f(x) then deg h(x) ≤ deg f(x). Therefore, degree of greatest divisor of f(x) is degree of f(x). ⇒Degree of greatest common divisor of f(x) and 0F is equal to degree of f(x). But, GCD(f(x), 0F) = 1⇒degree of greated common of f(x) and 0F is equal to degree of 1 which is 0. ⇒deg f(x) = 0 ⇒f(x) is constant polynomial. Therefore, f(x) must be constant polynomial. Answer: If f(x) and 0F are relatively prime then f(x) is constant.