,
.
,
.
.
, .
III. 11.
`.
p a p , , , .
, ` 1, 2, 3,.., 10 .
,
(mod 11).
1, 3, 4, 5, 9 11.
IV. a = 2 m = 15.
`.
a m, :
1.) (a, m)=1;
2.) - ;
3.) - k, .
(2, 15)=1; :
= .
, 1, 2, 4, 8. :
,
,
.
, =4.
V. m =7.
`.
m =7 . :
.
m -1=6 , , . .
. 2.
,
.
3<6, 2 7.
. , 3 6, 3 7.
, (k, m -1)=(k, 6)=1 1< k <6/
ֳ k =5. , . , 7 3 5.
VI. ` . (1)
`.
.
:
;
,
(2)
ij ind x. ` . (18, 22)=2 4 2, 2 `.
2:
.
11:
.
9:
.
ij ` (2):
,
.
x:
, .
VII. x, .
`.
:
.
:
;
,
.
, k=0, 1, 2,..
VIII. 35.
`.
.
; (13, 35)=(12, 35) = 1, .
.
, 35 11.
9
. f(x) g(x) m(x) n(x), f(x)m(x) + g(x)n(x) = d(x).
1) f(x) = x 4 + x 3 3 x 2 4 x 1; g(x) = x 3 + x 2 x 1;
2) f(x) = x 6 7 x 4 + 8 x 3 7 x + 7; g(x) = 3 x 5 7 x 3 + 3 x 2 7;
3) f(x) = x 5 + x 4 x 3 3 x 2 3 x 1; g(x) = x 4 2 x 3 x 2 2 x + 1;
4) f(x) = x 4 10 x 2 +1; g(x) = x 4 4 x 3 + 6 x 2 + 4 x + 1;
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5) f(x) = x 5 + 3 x 4 + x 3 + x 2 + 3 x + 1; g(x) = x 4 + 2 x 3 + x + 2;
6) f(x) = 4 x 4 2 x 3 16 x 2 + 5 x + 9; g(x) = 2 x 3 x 2 4 x + 4;
7) f(x) = x 4 x 3 4 x 2 + 4 x + 1; g(x) = x 2 x 1;
8) f(x) = x 5 5 x 4 2 x 3 + 12 x 2 2 x + 12; g(x) = x 3 5 x 2 3 x + 17;
9) f(x) = 3 x 4 3 x 3 + 4 x 2 x + 1; g(x) = 2 x 3 x 2 + x + 1;
10) f(x) = x 4 + x 3 + x 2 + x + 1; g(x) = 4 x 3 + 3 x 2 + 2 x + 1.
. :
) f(x) ( ) ;
) ( );
) f(x) = .
1) f(x) = x 4 + 3 x 3 8 x 2 + 4 x 1; a = 2;
2) f(x) = x 5 + 3 x 4 9 x 3 7 x 2 + 39 x 21; a = 1;
3) f(x) = x 4 2 x 3 5 x 2 + 2 x + 2; a = 2;
4) f(x) = x 6 6 x 4 4 x 3 + 9 x 2 + 12 x + 2; a = 3;
5) f(x) = x 5 10 x 3 20 x 2 15 x 4; a = 1;
6) f(x) = x 5 6 x 4 + 16 x 3 24 x 2 + 20 x 8; a = 3;
7) f(x) = x 6 2 x 5 x 4 2 x 3 + 5 x 2 + 4 x + 4; a = 1;
8) f(x) = x 6 15 x 4 + 8 x 3 + 51 x 2 72 x + 27; a = 1;
9) f(x) = x 7 3 x 6 + 5 x 5 7 x 4 + 7 x 3 5 x 2 + 3 x 1; a = 2;
10) f(x) = 3 x 4 + 6 x 3 2 x 2 + 1; a = 1.
. .
1) f(x) = x 4 2 x 3 8 x 2 + 13 x 24;
2) f(x) = 6 x 4 + 19 x 3 7 x 2 26 x + 12;
3) f(x) = x 5 2 x 4 4 x 3 + 4 x 2 5 x + 6;
4) f(x) = 10 x 4 13 x 3 + 15 x 2 18 x 24;
5) f(x) = x 4 + 2 x 3 13 x 2 38 x 24;
6) f(x) = x 4 + 4 x 3 2 x 2 12 x + 9;
7) f(x) = x 5 + x 4 6 x 3 14 x 2 11 x 3;
8) f(x) = 2 x 3 + 3 x 2 + 6 x 4;
9) f(x) = 2 x 3 3 x 2 + 4 x 5;
10) f(x) = x 4 x 3 22 x 2 + 16 x + 96.
V. :
1) f (x 1, x 2, x 3) = x 13 + x 23 + x 33 x 1 x 2 x 3;
2) f (x 1, x 2, x 3) = x 15 x 2 x 3 + x 25 x 1 x 3 + x 35 x 1 x 2 + 2 x 1 x 2 x 3;
3) f (x 1, x 2, x 3) = x 14 x 22 + x 24 x 12 + x 34 x 22 + x 34 x 12 + x 14 x 32 + x 24 x 32;
4) f (x 1, x 2, x 3) = x 12 x 2 + x 1 x 22 + x 12 x 3 + x 1 x 32 + x 22 x 3 + x 2 x 32;
5) f (x 1, x 2, x 3) = x 14 + x 24 + x 34 x 13 2 x 12 x 22 2 x 22 x 32 2 x 12 x 32;
6) f (x 1, x 2, x 3) = (x 1 x 2)2 + (x 1 x 3)2 + (x 2 x 3)2;
7) f (x 1, x 2, x 3) = (x 1 + x 2 5 x 3) (x 2 + x 3 5 x 1) (x 1 + x 3 5 x 2);
8) f (x 1, x 2, x 3) = 3 x 13 3 x 23 + 3 x 33 + x 1 + x 2 + x 3;
9) f (x 1, x 2, x 3) = 3 x 13 + 3 x 23 + 3 x 33 + 5 x 1 x 2 x 3 + 2 x 12 + 2 x 22 + 2 x 32;
10) f (x 1, x 2, x 3) = (x 1 x 2)(x 2 x 3)(x 3 x 1).
V. :
1) 2) 3)
4) 5)
6) 7)
8) 9)
10)
VI. :
1) 2) 3)
4) 5)
6) 7) 8)
9) 10)
V. , a Q , :
1) | 6) |
2) | 7) |
3) | 8) |
4) | 9) |
5) | 10) |
V. Q :
1) f(x) = x 4 + x 3 6 x 2 7 x 7;
2) f(x) = x 4 x 3 6 x 2 + 8 x 2;
3) f(x) = 6 x 4 13 x 3 + 12 x 2 13 x + 6;
4) f(x) = 9 x 4 15 x 3 + 28 x 2 20 x + 16;
5) f(x) = (x + 3)4 + (x + 5)4 16;
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6) f(x) = (x + 1)6 9(x + 1)3 + 20;
7) f(x) = x 4 + 4 x 3 + 4 x 2 + 1;
8) f(x) = x 3 6 x 2 + 11 x 6;
9) f(x) = x 4 + 3 x 3 3 x 2 11 x 6;
10) f(x) = x 5 x 3 x 2 + 1.
1. | .1, .1, . 1, V.1, V.1, V.1, V.1, V.1 |
2. | .2, .2, .2, V.2, V.2, V.2, V.2, V.2 |
3. | .3, .3, .3, V.3, V.3, V.3, V.3, V.3 |
4. | .4, .4, .4, V.4, V.4, V.4, V.4, V.4 |
5. | .5, .5, .5, V.5, V.5, V.5, V.5, V.5 |
6. | .6, .6, .6, V.6, V.6, V.6, V.6, V.6 |
7. | .7, .7, .7, V.7, V.7, V.7, V.7, V.7 |
8. | .8, .8, .8, V.8, V.8, V.8, V.8, V.8 |
9. | .9, .9, .9, V.9, V.9, V.9, V.9, V.9 |
10. | .10, .10, .10, V.10, V.10, V.10, V.10, V.10 |
9
1. f(x) g(x) m(x) n(x), f(x)m(x) + g(x)n(x) = d(x).
f(x) = 2 x 4 + 3 x 3 3 x 2 5 x + 2;
g(x) = 2 x 3 + x 2 x 1.
.
f(x) g(x) :
_ 2 x 4 + 3 x 3 3 x 2 5 x + 2 | 2 x 3 + x 2 x 1 | ||||||||
2 x 4 + x 3 x 2 x | + 1 | ||||||||
_ 2 x 3 2 x 2 4 x + 2 | |||||||||
2 x 3 + x 2 x 1 | |||||||||
_ 2 x 3 + x 2 x 1 | 3 x 2 3 x + 3 | ||||||||
2 x 3 + 2 x 2 2 x | |||||||||
_ x 2 + x 1 | |||||||||
x 2 x + 1 | |||||||||
_ 3 x 2 3 x + 3 | 2 2 | ||||||||
3 x 2 + 3 x | |||||||||
_ 6 + 3 | |||||||||
6 + 6 | |||||||||
3. | |||||||||
, :
f(x) = g(x)q 1 (x) + r 1 (x);
q 1 (x) = x + 1, r 1 (x) = 3 x 2 3 x + 3;
g(x) = r 1 (x)q 2 (x) + r 2 (x);
q 2 (x) = ; r 2 (x) = 2 x 2;
r 1 (x) = r 2 (x)q 3 (x) + r 3 (x);
q 3 (x) = ; r 3 (x) = 3.
r 3 (x) = 3 , - , r 4 (x) . , , 3,
d(x) = 1 = r 3 (x).
d(x) m(x) n(x) r 3 (x).
r 3 (x) = r 1 (x) r 2 (x) q 3 (x),
r 3 (x) = r 1 (x) [ g(x) r 1 (x)q 2 (x) ] q 3 (x),
r 3 (x) = r 1 (x) [1 + q 2 (x)q 3 (x) ] g(x)q 3 (x).
r 1 (x) , :
r 3 (x) = [ f(x) g(x)q 1 (x) ]×[1 + q 2 (x)q 3 (x) ] g(x)q 3 (x) =
= f(x) [1 + q 2 (x)q 3 (x) ] + g(x) [ q 1 (x) q 1 (x)q 2 (x)q 3 (x) q 3 (x) ].
, d(x) = r 3 (x), :
.
, ;
,
q 1 (x) = x + 1; q 2 (x) = ; q 3 (x) = .