7.)a. Prove that = {a, + apx + *. + ar" |a, = 2k for k E Z), %3D %3D the set of all polynomials in Zx]with even constant term, is an ideal of Z[x). b. Show that /[x] = {x•f(x) + 2 g(x)| f(x), g(x) E...

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Extracted text: 7.)a. Prove that = {a, + apx + *. + ar" |a, = 2k for k E Z), %3D %3D the set of all polynomials in Zx]with even constant term, is an ideal of Z[x). b. Show that /[x] = {x•f(x) + 2 g(x)| f(x), g(x) E Z]}. 8. a. Prove that I[x] = {do + a* + ** + a"|a, 2k, for k,E Z), 2k, for k, E Z), %3D %3D the set of all polynomials with even coefficients, is an ideal of Z[x. b. Prove or disprove that /Ex is a principal ideal. 9. Ja. Let F be a field. Prove that I[x] = {ao + ax +*** + a,r"|a, E Fand a, + a, + + a, = 0}, the set of all polynomials in F[x] such that the sum of the coefficients is zero, is an ideal of F[x]. ... %3D %3D b. Prove or disprove that /[x] is a principal ideal.