xx（1）设多项式f(x)=
1x2x013x2343，则多项式的次数为（B） xx（A）2 （B）3 （C）4 （D）5 解：方法一：
xxf?x??1x2x013x2343xx1??r1?r3??x?xx0x212x33x34x1r2?xr10?r3?xr10r4?xr1002xx?x3?x223?2x4?x213?2xx?x2??r2?2r1??r3?xr1按第一列展开x?x3?x2?23?2x4?x213?2xx?x2??r1?r3??13?2xx?x223?2x4?x2x?x3?x213?2xx?x202x?3x2?2x?40?x?4?2x?x3?2x2?3
按第一列展开2x?3x2?2x?432?x?10x?22x?9?x?4?2x?x3?2x2?3?多项式次数为3；
方法二：
x23xxxf(x)?102x13??r3?r1?43xx??c3?2c1??c4?xc1x23?2x4?x2xx?x3?x21000x13?2xx?x2??c3??x?4?c1??23?2x4?x2按第三行展开3?11???1??x?x3?x213?2xx?x223?2x4?x2x?x3?x2?10x?423?2x?x2?2x?4x?x?4x?3?100按第三行展开??1????1?3?13?2x?x2?2x?4??x3?10x2?22x?9?x?4x?3?多项式次数为3；
注意：实际上方法一与方法二思想类似：利用行列式展开定理对行列式降阶，最后求出行列
式的值（多项式）.
xx方法三：f(x)?1x2x013x2343xx按第二行展开xA21?xA22?xA23?3A24
这四项的最高次项分别为：x2，x3，x3，x
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234A21???1?02x?O?x?
13xx34A22?12x?2x2?3x2?12??8x?3x?3x2??2x2?O?x?
x3xx24A23??10x122???x2?O?x? x???0?2x?4?0?2x?x????x?f(x)?xA22?xA23?O?x2??2x3?x3?O?x2??x3?O?x2?
?多项式次数为3.
x（2）设x,y为实数且?yy0x0=0，则（D） x10（A）x?0,y?1 （B）x??1,y?1 （C）x?1,y??1（D）x?0,y?0
x解：?yy0x0?x2?y2?0?x?y?0. x10a11?xa12?xa13?xa14?xa21?xa22?xa23?xa24?x（3）设多项式f(x)=，则多项式的次数最多为（A）
a31?xa32?xa33?xa34?xa41?xa42?xa43?xa44?x（A）1 （B）2 （C）3 （D）4
?1???1??解：设1?，A??aij????1?2?3?4?，则
4?4?????1?f(x)??1?x1?2?x1?3?x1?4?x1性质4?1?2?x1?3?x1?4?x1?x1?2?x1?3?x1?4?x1
?f1?x??f2?x?f1?x???1?2?x1?3?x1?4?x1
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性质4?1?2?3?x1?4?x1??1x1?3?x1?4?x1
?f3?x??f4?x?f2?x?性质3x1?2?x1?3?x1?4?x1性质4性质5x1?2?3?4?O?x?
x1?4?x1f3?x???1?2?3?x1?4?x1?f5?x??f6?x??1?2?3?4?x1??1?2f4?x?性质3x?11?3?x1?4?x1性质4性质5x?11?3?4?O?x?
f5?x???1?2?3?4?x1性质3?1?2?3?4??1?2?3x1
?1?2?3?4?x?1?2?31?O?x?性质5f6?x?性质3x?1?21?4?x1x?1?21?4?O?x?
?f(x)?f1?x??f2?x???f3?x??f4?x???f2?x?????f5?x??f6?x???f4?x????f2?x??O?x??f(x)的次数最多为1.
0（4）Dn??1?1?，当n=（ C ）时，Dn?0.
?1?1解：Dn?0（A）3 （B）4 （C）5 （D）7
?1?1???1?n?nn?n?1?2???1????1?nn?n?1?2
n3457 n?n?1?61015282?当n?5时，Dn??1?0，选C.
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（5）?j为四阶行列式D的第j列，（j=1，2，3，4，），且D=?5，则下列行列式中，等于?10的是（D）. （A）2?1,2?2,2?3,2?4
（B）?1??2,?2??3,?3??4,?4??1
（C）?1,?1??2,?1??2??3,?1??2??3??4
（D）?1??2,?2??3,?3??4,?4??1
解：D??1?2?3?4??5
（A）D1?2?12?22?32?4?24?1?2?3?4?24D??80
（B）方法一：D2??1??2?2??3?3??4?4??1
c1?c2?c3?c40?2??3?3??4?4??1?0
方法二：D2??1??2?2??3?3??4?4??1
性质4?1?2??3?3??4?4??1??2?2??3?3??4?E1?F1c4?c1E1c3?c4?1?2?3?4?D
c2?c3c2?c1F31c3?c2?2?3?4?1???1???1?2?3?4??D
c4?c3?D2?E1?F1?D?D?0
（C）D3??1?1??2?1??2??3?1??2??3??4
c4?c3?c3?c21?1??2?1??2??3?4?1?1??2?3?4c2?c1?1?2?3?4?D??5（D）D4??1??2?2??3?3??4?4??1
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4??1
?
性质4?1?2??3?3??4?4??1??2?2??3?3??4?4??1?A1?B1c4?c1A1c3?c4?1?2?3?4?D
c2?c3c2?c1B3?11c3?c2?2?3?4??1???1???1?2?3?4?D
c4?c3?D4?A1?B1?D?D?2D??10
3．计算下列行列式
11114124（1）
123413610， （2）120210520，
1410200?1?1?7311126113（3）
1311102041131， （4）21350 ， 111313410303690000230000561?abcd（5）
100800a1?230000 ， （6）bcdab1?cd，
045000abc1?d006700abacae1111（7）bd?cdde， （8）abcd2bfcf?efab2c2d2. a4b4c4d4111111111111134r4?r10123r4?3r20123r4?3r解：（1）D?123013610r3?r1r0259r3?2r2001301410202?r10391900310014
111123013?1001