高等代数 多项式

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1、koeijzlpwt

2、mrq{nujgfvux(hysei)b_W^1.242-E(x−1)

3、Ax+Bx+1?A=,B=.2.Q5Z9?Q&(5A.3.9P~(,/!x+a8f(x)HÆ(*f(−a).4.23

4、(x−1)

5、ax+bx+2,?a,b6.5.f(x)=x3+2x2+6x+5,g(x)=x3−3x−2,?(f(x),g(x))=.6.f(x),g(x)5Z/!tf(x),g(x)>99n?>?f(x),g(x)>5Z9~(?2.7.k

6、Y;/!p(x)f(x)(s2

7、p(x)

8、f(x)mep(x)f(x)(s+k28.f(x)=x4+x3−3x2−4x−1,g(x)=x3+x2−x−1,(?2(f(x),g(x))=.TZS^q-`<

9、Cyu=Dj

10、uX31.{D5Z9~/!f(x)>5Z9~>?f(x)>5Z9~Y;2.q-`/!>5Z9~8Y;jZ4332f(x)=x+6x+1;g(x)=x+6x+6x+1.3.f(x),g(x),d(x),9P~(/!{D>/!u(x),v(x),'d(x)=f(x)u(x)+g(x)v(x),

11、?d(x)f(x)8g(x)(?21′4.f(x)9P~(/!f(x)f(x)(%Bk≥1.{D′Y;/!p(x)f(x)(k2?p(x)f(x)(k+12[a^1.F906=f(x)∈F[x],p(x)F[x]Y;/!{D>9c0,'f(c0)=p(c0),?.(A)f(x)=g(x)(B)f(x)

12、p(x)(C)f(x)=ap(x),a6=0(D)p(x)

13、f(x)2.9~Y;(/!f(x).(A)x2+3x+3(B)x3+3x+3(C)x2−3x−3(D)x3−3x−3

14、U℄^1.3va,bd('/!x+3ax+b522.=_n;(a1,a2,···,an,v,;(/!f(x)f(ai)=bi,[bi+=+(Gi=1,2,···,n.3.3va,bd('/!x+3ax+b524.2v,;}/!f(x)'f(x)+1o(x−1)B0f(x)−12o(x+1)B`dY^1.33F9f,g∈F[x].{Dg

15、f,?g

16、f.2.F,F19tF⊆F1,f(x),g(x)∈F[x].(1)Dj{D>F1[x]5g(x)

17、f(x),?>

18、F[x],+5g(x)

19、f(x).(2)Djf(x)8g(x)>F[x]HÆ$tU$f(x)8g(x)>F1[x]HÆ(3)Djf(x)9F~(Y;/!?f(x)(>,">3.dnDjx−1

20、x−1$tU$d

21、n.,x,···,x)=x4+x4+···+x44../!f(x12n12nÆ)./2!(/!5./!f(x),g(x)HÆ(Dj>℄;/!a(x),b(x),'a(x)f(x)+b(x)g(x)=1.6.Fz/9f(x)F~(,:/!r!a,′n′n.Djf(

22、x)

23、f(x)$tU$>b∈F'f(x)=a(x−b),rf(x)Æ′′f(x)(%f(x)

24、f(x)Æf(x)Bf(x).ab6=0.7.f1(x)=af(x)+bg(x),g1(x)=cf(x)+dg(x)tcdDj(f(x),g(x))=(f1(x),g1(x)).x2x3xp8.Dj/!f(x)=1+x+++···+>5Z9~Y;p2!3!p!Æ9.(1))DjB6B/!Y;&(Eisensteinq2(2)q25l#CV^/()F10.44v7/!f(x),'(x−1)

25、f(

26、x)+10(x+1)

27、f(x)−1.11.5Dj/!f(x)=x−5x+1>5Z9Q~Y;12.3444α,β,γ34x+3x−1=0(>vα+β+γ(G13.3v/!f=x+px−q(qD(f)(3f(ÆD(f).q222(+0D(f)=(x1−x2)(x1−x3)(x2−x3),x1,x2,x3f(9>p,q)14.YouareencourgedtoanswerthefollowingquestioninEnglish.Letα1,α2,···,αnbedifferentintegers.Pr

28、ovethatthepolynomialf(x)=(x−α1)(x−α2)···(x−αn)+1isirreducibleorasquareofsomepolynomialove