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, ( 1758 ), 31 1730 27 1783 .

1763 , 1768 .

, . , , ( . ) , m n ⠠ mn . 1848 , 1764-69 . , , . . .

 

 

 

 

ࠠ .

 

Pn(x)

(x-a)

x = a.


:

Pn(x) n ,

(x-a) - ,

Qn-1(x) Pn(x) x-a ( n-1 ) ,

R ( R x x ).


:


:

Pn (x) = (x-a)Qn-1(x) + R .

x = a :

Pn (a) = (a-a)Qn-1 (a) + R =0*Qn-1(a)+R=

=0+R=R .

, R = Pn (a) , .. ࠠ ࠠ (x-a)

ࠠ 蠠 x=a ,  蠠 .



 

 

 

.

 


1 :

Pn (x)

ax+b

 ࠠ 蠠 x = -b/a ,

. . R=Pn (-b/a) .

 

:


:

Pn (x)= (ax + b)* Qn-1 (x) + R .

x= -b/a :

Pn (-b/a) = (a(-b/a) + b)Qn-1(-b/a) + R = R. , R = Pn (-b/a) , .


2:

蠠  a

P (x) ,

ࠠ (x-a)

.

 

:


P (x)x-a P (a) , a P (x) , , P (a) = 0 , .

, ࠠ P (x) = 0 ࠠ ࠠ P , ( ) .



3 :

P (x)

 堠

a1 , a2 , , an ,

堠 (x-a1) (x-an)

.

:

. 蠠 n=1 2 . , k , , P(x) 砠 ࠠ ࠠ (x-a1)(x-a2) (x-ak) ,

a1 , a2 , , ak - .

P(x) 򠠠 k+1 . a1 , a2 , ak , , ak+1 , , , (x-a1) (x-ak) , , 

P(x) = (x-a1) (x-ak)Q(x).

ak+1P(x) , . . P(ak+1) = 0 .

,  x ak+1 , :

P(ak+1) = (ak+1-a1) (ak+1-ak)Q(ak+1) =

=0 .

ak+1a1 , , ak , ak+1-a1 , , ak+1-ak 0 . ,  Q(ak+1) , . . ak+1Q(x) . 2 ,  Q(x)x-ak+1 .

Q(x) = (x-ak+1)Q1(x) , 󠠠

P(x) = (x-a1) (x-ak)Q(x) =

=(x-a1) (x-ak)(x-ak+1)Q1(x) .

 蠠 ,  P(x)(x-a1) (x-ak+1) .

, , 蠠 k =1 , 帠 蠠 蠠 n = k ,  n = k+1. , 蠠 堠 , .




4 :

n

n .

:


: Pn(x)n n 頠 - n+k (a1 , a2 , , an+k -  ) , 󠠠 3

(x-a1) (x-an+k) , 堠 n+k , .

, 蠠 n , 젠 n ,  .




5 :

 ࠠ P(x)

蠠 ࠠ a

(P(x)-P(a)) 砠

ࠠ (x-a) .

:


P(x) n , a - .

Pn(x)  ⠠ : Pn(x)=(x-a)Qn-1(x)+R ,

Qn-1(x) , 蠠 Pn(x) (x-a) ,

RPn(x)(x-a) .

:

R = Pn(a) , ..

Pn(x)=(x-a)Qn-1(x)+Pn(a) .

Pn(x) - Pn(a) = (x-a)Qn-1(x) ,

( Pn(x) Pn(a) )

(x-a) ,  蠠 .




6 :

a

P(x)

堠 堠 頠 ࠠ

 ,

P(x) ࠠ (x-a)

.

:


 .

1. .


a P(x) , 2 P(x)(x-a) 砠 .

젠 젠 P(x)(x-a) 젠 젠 , a P(x) , .. .


 

2. .


P(x)(x-a),

R = 0 , RP(x) (x-a) , R = P(a) , ,  P(a) = 0 , ,  a P(x) .

P(x)(x-a) , aP(x) .


P(x) (x-a) 젠 , a P(x) , .

7():

, 堠 頠 -

, ⠠

.


:


: - , P(x) 蠠 ࠠ (x a):

P(x) = (x a)Q(x),

(x a) , 6 aP(x) , . , ,

頠 , , .



ࠠ 蠠 5  :


1. :


P(x) = xn , P(a) = an ,

xn an .

5

P(x) - P(a) = xn an = (x a)Q(x) ,

,

(xnan)/(xa)=Q(x), .. ࠠ ,  蠠 .

(xn an)/(x a) = xn-1 + axn-2 + a2xn-3 + +an-2x + an-1.



2. .


P(x) = x2k , P(a) = a2k .

x2k - a2kP(x) P(a) .

P(a) = a2k = (-a)2k = P(-a) , .. x2k - a2k = P(x) P(-a).

5

P(x) - P(-a) = (x (- a))Q(x)=

= (x + a)Q(x)

,

x2k a2k = (x + a)Q(x)

(x2k a2k)/(x + a) = Q(x) ,

.. 頠 ,  蠠 .

,

(x2k a2k)/(x + a) = x2k-1 ax2k-2 + +a2k-2x + a2k-1.

3. .


P(x) = x2k+1 - a2k+1 .

 󠠠 蠠 蠠 x2k+1 - a2k+1 x + a = x (-a)

R = P(-a) = (-a)2k+1 a2k+1 = -2a2k+1

. . 0 , , 蠠 .



4. 頠 .

P(x) = x2+1 , P(-a) = (-a)2+1 = -2+1 ,

P(x) P(-a) = x2k+1 + a2k+1 .

5

P(x) - P(-a) = x2k+1 + a2k+1= (x (- a))Q(x)=

= (x + a)Q(x),

,

(x2k+1 + a2k+1)/(x + a) = Q(x) ,

.. 頠 ,  蠠 .

,

(x2k+1 + a2k+1)/(x + a) = x2k - ax2k-1 + - a2k-1x + a2k.

5. .


P(x) = x2k + a2k .

 󠠠 蠠 蠠 x2k + a2k x + a = x (-a)

R = P(-a) = (-a)2k + a2k = 2a2k.

. . 0 ,

, .

⠠ .



1.

x3 3x2 + 6x 5

x 2 .


R = P3 (2) = 23 3*22 + 6*2 5 = 3 .

: R = 3 .


2.

32x4 64x3 + 8x2 + 36x + 4

2x 1 .


1

R=P4(1/2)=32*1/2464*1/23 + 8*1/22+36*1/2+4=

= 2 8 + 2 + 18 + 4 =18 .

: R = 18 .



3.

a

x4 + ax3 + 3x2 4x 4

x 2 ?


R = P4 (2) = 16 + 8a + 12 8 4 = 8a +16.

R = 0 ,

8a + 16 = 0 ,

a = -2 .

: a = -2 .

4.

ab

ax3 + bx2 73x + 102

x2 5x + 6 ?


:

x2 5x + 6 = (x 2)(x 3) .

x 2 x 3 , 頠 ࠠ x 2 蠠 ࠠ x 3 , ࠠ  ,

R1 = P3 (2) = 8a + 4b 146 + 102 =

= 8a + 4b 44 = 0

R2 = P3 (3) = 27a+9b 219 + 102 =

= 27a +9b -117 =0


:


8a + 4b 44 = 0

27a + 9b 117 = 0

2a + b = 11

3a + b = 13


:

a = 2 , b = 7 .


: a = 2 , b = 7 .

5.


a b

x4 + ax3 9x2 + 11x + b

x2 2x + 1 ?


:

x2 2x + 1 = (x 1)2

x 1 ,


R1 = P4 (1) = 1 + a 9 + 11 + b = a + b + 3 = 0.

x 1 :

_ x4 + ax39x2 + 11xa 3 x 1

x4 x3 x3+(a+1)x2+(a8)x+(a+3)

_(a + 1)x3 9x2

(a + 1)x3 (a + 1)x2

_(a 8)x2 + 11x

(a 8)x2 (a 8)x

_(a + 3)x a 3

(a + 3)x a 3

0

x3+(a+1)x2+(a8)x+(a+3)

(x 1) ,

R2 = P3 (1) = 1 + (a + 1)*1 +(a 8)*1 + a+3 =

=3a 3 = 0 .

a + b + 3 = 0

3a 3 = 0


a + b =-3

a = 1

: a = 1 , b = -4

: a = 1 , b = -4 .


6.

P(x) = x4 + 4x2 5 .

1P(x) , ,  2 P(x) (x 1) :



_x4 + 4x2 5 x 1

x4 x3 x3 + x2 + 5x + 5

_x3 + 4x2 5

x3 x2

_5x2 5

5x2 5x

_5x 5

5x 5

0



P(x)/(x 1) = x3 + x2 + 5x + 5 ,

P(x) = (x 1)(x3 + x2 + 5x + 5).


x3 + x2 + 5x + 5 x = -1 , ,  2 󠠠 x3 + x2 + 5x + 5 (x + 1) :


_x3 + x2 +5x + 5 x + 1

x3 + x2 x2 +5

_5x + 5

5x + 5

0

(x3 + x2 +5x + 5)/(x + 1) = x2 +5 ,

x3 + x2 +5x + 5 = (x +1)(x2 +5).

P(x) = (x 1)(x +1)(x2 +5) .


7 (x2 + 5) , .. , P(x) .


: x4 + 4x2 5 = (x 1)(x +1)(x2 +5) .


7.

P(x) = x4 + 324 .


P(x) , .. x4 -324 , , 7 P(x) .


: .




8.

2

P(x) = x5 - 5x4 + 7x3 2x2 + 4x 8 .


 

: P(x) (x a)k , ࠠ (x a)k+1 , , a k P(x).


_x5 - 5x4 + 7x3 2x2 + 4x 8 x 2

x5 - 2x4 x4 3x3 + x2 + 4

_-3x4 + 7x3 2x2 + 4x 8

-3x4 + 6x3

_x3 2x2 + 4x 8

x3 2x2

_4x 8

4x 8

0

_x4 3x3 + x2 + 4 x 2

x4 2x3 x3 x2 x 2

_-x3 + x2 + 4

-x3 +2x2

_-x2 + 4

-x2 + 2x

_-2x + 4

-2x + 4

0

_ x3 x2 x 2 x 2

x3 2x2 x2 + x + 1

_x2 x 2

x2 2x

_x 2

x 2

0

x2 + x + 1 x 2 , .. R=22 + 2 + 1=

=7.

, P(x)/(x 2)3 = x2 + x + 1 , .. 2 3 P(x) .


: 2 3 P(x) .


9.

, 4 2 -2 .


3 , P(x) 4 2 2 , ࠠ (x 4)2(x + 2) ,

P(x)/(x 4)2(x + 2) = Q(x) ,

.. P(x) = (x 4)2(x + 2)Q(x) =

= (x2 8x +16)(x + 2)Q(x) =

= (x3 8x2 + 16x +2x2 16x + 32)Q(x) =

= (x3 6x2 + 32)Q(x).


(x3 6x2 + 32) - , P(x) , , Q(x) .

Q(x) = 1 , P(x) = x3 6x2 + 32 .


: x3 6x2 + 32 .




10.

a b , -2P(x) = x5 + ax2 + bx + 1, .


-2 P(x) , 3 P(x) (x + 2)2 (R = 0)

(x + 2)2 = x2 + 4x + 4


_x5 + ax2 + bx + 1 x2 + 4x + 4

x5 + 4x4 + 4x3 x3 4x2 + 12x (a + 32)

_-4x44x3ax2+bx+1

-4x4 16x3 16x2

_12x3 + (16 a)x2 + bx + 1

12x3 +48x2 + 48x

_-(a + 32)x2 + (b 48)x + 1

-(a + 32)x2 4(a + 32)x 4(a + 32)

(4a +b 48 + 128)x + 4a + 129

R = (4a +b 48 + 128)x + 4a + 129 =

= (4a +b + 80)x + 4a + 129

R = 0 ,

(4a +b + 80)x + 4a + 129 = 0 x .

,

4a +b + 80 = 0 ,

4a + 129 = 0

:


4a +b + 80 = 0 a = -32,25

4a + 129 = 0 b = 49

: a = -32,25 , b = 49 .



, ࠠ 堠 蠠 蠠 , ⠠ ( , .. ) , 젠 , 젠 .

蠠 蠠 .




 

 

 

 

.


1. .., ..

蠠 .

2. .


3. .., ..

.


4. .., - .., - ..

.

5. ..

.