多项式习题大全

一.填空.

(1)3()105f x x x =-+,则()f x 在有理数域上_______ (可约,不可约),()f x 在实数域上有_________个根.

(2)已知320x px qx r +++=的三个根是123,,x x x ,求一个三次方程使其根为222123

,,x x x . (3)4322()441,()1f x x x x x g x x x =--++=--,则存在()_______u x =,()_____v x =,d(x)=______,使()()()()()d x f x u x g x v x =+.

(3)当整数______a =时,多项式32122

x x ax -+-有有理根. (4)多项式43242203x x x x -+++有一根32i +,则其余根__________.

(5)________.

(6)已知432()410129f x x x x x =++++有重根,则((),())______f x f x '=,()f x 的所有根是________.

(7)4()()f x x p x q =++,则()f x 的判别式_________.

(8)设323210()f x a x a x a x a =+++,则()f x 被x c -除所得的商式为_________,余式为_______.

(9)当l 与m 满足_______时,2321|52x mx x lx x +++++.

(10)将5432()252243f x x x x x x =+-+--表示成3x -的方幂的和__________.

(11)求一个2次多项式,使它在0,

,2x ∏=∏处与函数sin x 有相同的值________. (12)多项式3x px q ++有重根的条件________.

(13)5432

()252243f x x x x x x =+-+--,在实数域上的标准分解_______;在复数域的标准分解_________.

(14)设2()4g x x x a =-+,且存在唯一一个首一的3次多项式()f x ,使得()|()g x f x ,且2()|()f x g x ,则______a =,()_______f x =.

(15)若多项式32521x x x +-+三根123,,ααα,则333123________ααα++=.

(18)543222x x x x x +-+--的有理根是________,在有理数域上的标准分解式为________.

(19)设4322()21,()21f x x x x x g x x x =--+-=-+,则存在()_u x =,()_______v x =,使得()()()()((),())u x f x v x g x f x g x +=.

(21)i 为根的次数最小的有理系数多项式是_________.

(22)已知实系数多项式3x px q ++有一个虚根32i +,则其余两个根是_________.

(23)关于整系数多项式不可约性的Eisenstein 判别法是________.

二.选择.

(1)下列命题中正确的有_____个.

()(),()[],()I p x f x X p x ∈K 是不可约多项式,则()|()()p x f x p x 与

()f x 在 上有公共根.

()II ((),())1(),()f x g x f x g x = 无公共根.

()III ()f x 是三次有理多项式,则()f x 在有理数域上不可约.

()IV 如果((),())1f x f x '''=,则()f x 的重因式是2重因式.

A.1

B.2

C.3

D.4

(2)下列命题中错误的有______个.

()I 若12()()(),()()()f x af x bg x f x cf x dg x =+=+,则11((),())((),())f x g x f x g x =. ()II 若()0,()0,()()()()((),())f x g x f x p x g x q x f x g x ≠≠+=,则((),())1p x q x =. ()III 若(()(),())1f x g x h x =,则((),())1,((),())1f x h x g x h x ==.

()IV 若()|()()f x g x h x 且()|()f x g x ,则((),())1f x h x =.

A.1

B.2

C.3

D.4

(3)如果实系数多项式10110()n n n f x a x a x a x a --=++++ 系数满足011210,0,,0n n a a a a a a -<<< ,则()f x 的根______.

A.0>

B.0>

C.0=

D.0≤

(4)已知α是()f x 与(1)()k f x -的根,但不是()()k f x 的根,则α______.

A.是()f x 的k 重根.

B.是()f x 的单根.

C.不是()f x 根.

D.未必是()f x 的k 重根.

三.判定题(若命题成立,给出证明;若命题不成立,给出反例)(24分,每题6分).

(1)设()|()()f x g x h x ,则或()|(),f x g x 或()|()f x h x .

(2)设()f x 和()g x 是实数域上的多项式且()f x 和()g x 在复数域上无公共根,则((),())1f x g x =.

(3)设()f x 是有理数域上的多项式,则()f x 在有理数域上无重因式当且仅当9)f x 在复数域上无重根.

(4)设不可约多项式()p x 是()f x '的因式,则()p x 是()f x 的重因式.