˜A—§•û’öŽ®E•s“™Ž®‚Ì–â‘è
 

Author: Mathcot.H.I.

Original: 2007.07.01

Update: 2014.02.26



˜A—§•û’öŽ®

[‰‰K1]
[‰ð“š1]
[Maple10]@


[‰‰K2]@ ¶—“IH‰–…‚ð‚P‚O‚O‚O‚‡ì‚肽‚¢B…‰½ƒOƒ‰ƒ€‚ÉH‰–‰½ƒOƒ‰ƒ€‚𬂺‚ê‚Îì‚ê‚é‚©H
[‰ð“š2]
070604-004.gif


[‰‰K3]
ren1-2a.PNG

[‰‰K4]
ax+by=3, ax^2 +by^2=7, ax^3 +by^3=16, ax^4 +by^4=42
‚ð‰ð‚«A‚»‚ÌŽž‚Ì
ax^5 +by^5
‚Ì’l‚ð‹‚ß‚æB

[‰ð] [Maple10Žg—p]
solve([ax+by=3,ax^2+by^2=7,ax^3+by^3=16,{y,a,b})
{y=(7x-16)/(3x-7),a=-1/(x(3x^2-14x+16)),b=((3x-7)^3)/((7x-16)(3x^2-14x+16))}
solve(eval(ax^4+by^4=42,[y=(7x-16)/(3x-7),a=-1/(x(3x^2-14x+16)),b=((3x-7)^3)/((7x-16)(3x^2-14x+16))]),{x})
{x=-7+ã87},{x=-7-ã87}
c:=-7+ã87;d:=-7-ã87

{x=-7+ã87}
a:=factor(simplify(eval(-1/(x(3x^2-14x+16)),x=c)))
a:=(49/76)+(457/6612)ã87
b:=factor(simplify(eval(((3x-7)^3)/((7x-16)(3x^2-14x+16)),x=c)))
b:=(49/76)-(457/6612)ã87
y:=factor(simplify(eval((7x-16)/(3x-7),x=c)))
y:=-7-ã87

{x=-7-ã87}
a:=factor(simplify(eval(-1/(x(3x^2-14x+16)),x=d)))
a:=(49/76)-(457/6612)ã87
b:=factor(simplify(eval(((3x-7)^3)/((7x-16)(3x^2-14x+16)),x=d)))
b:=(49/76)+(457/6612)ã87
y:=factor(simplify(eval((7x-16)/(3x-7),x=d)))
y:=-7+ã87

factor(eval(ax^5+by^5,[a=(49/76)+(457/6612)ã87,(49/76)-(457/6612)ã87,x=-7+ã87, y=-7-ã87]))
20

ˆÈã‚©‚ç
(x,y,a,b)=(-7+ã87,-7-ã87,(4263+457ã87)/6612,(4263-457ã87)/6612),
(-7-ã87,-7+ã87,(4263-457ã87)/6612,(4263+457ã87)/6612)
ax^5+by^520

[•Ê‰ð]
ax+by=3, ax^2 +by^2=7, ax^3 +by^3=16, ax^4 +by^4=42 c(D)

(ax+by)(x+y)=ax^2+by^2+xy(a+b)
3(x+y)=7+xy(a+b)@c(A)

(ax^2+by^2)(x+y)=ax^3+by^3+xy(ax+by)
7(x+y)=16+3xy@c(B)

(ax^3+by^3)(x+y)=ax^4+by^4+xy(ax^2+by^2)
16(x+y)=42+7xy@c(C)
(B),(C)‚©‚ç
x+y=-14, xy=-38@c(E)
(A)‚É‘ã“ü
a+b=49/38@c(F)
x,y‚Í
z^2+14z-38=‚Ì2ª
x,y=-7}ã87@c(G)
(F),(G)‚ð(D)‚É‘ã“ü
-(7*49/38)+(a-b)ã87=3@c(H)
(F),(H)‚©‚ç
a,b=(49/76)}(457/6612)ã87@c(I)
‚Ü‚Æ‚ß‚Ä
(a,b,x,y)=((49/76)+(457/6612)ã87,(49/76)-(457/6612)ã87,-7+ã87,-7-ã87) or
((49/76)-(457/6612)ã87,(49/76)+(457/6612)ã87,-7-ã87,-7+ã87)

ax^5+by^5=(x+y)(ax^4+by^4)-xy(ax^3+by^3)=-14*42+38*16=20
tenn
s—ñ‚ð—p‚¢‚½˜A—§•û’öŽ®‚̉ð–@

[‰‰K1]
A@=  [ 1 2 ]@
@@   [ 1 1 ]

[X] = A [x]  ‚Æ‚¨‚­‚Æ
[Y]       [y]

[x] = A -1  [X] = [ -1  2 ] [X]
[y]            [Y]    [  1  -1] [Y]

‘‚«‰º‚·‚Æ
x=-X+2Y, y=X-Y

‚±‚ê‚ðuy = xv‚É‘ã“ü‚µ‚Ä

 -X+2Y=X-Y Ë 3Y = 2X

@ˆ Y = (2/3) X © (“š‚¦)


[‰‰K]
ŽŸ‚̘A—§ˆêŽŸ•û’öŽ®‚ð‰ð‚«‚È‚³‚¢
5x + 3y - 3z = 2
2x - y + z = 3
x + y + z = 6

[‰ð“š]

ŒW”s—ñ‚Í

       [ 5  3 -3 ]                              
A = [ 2 -1 1  ] ,  det A = -22,
       [ 1  1  1 ]                              

            [ -2 -6    0  ]                [2   6  0  ]
A -1 = [ -1  8  -11 ] / det A = [1 -8 11 ] /22
            [ 3  -2  -11 ]                [-3 2 11 ]

˜A—§ˆêŽŸ•û’öŽ®‚ð‘‚«Š·‚¦‚é‚Æ

    [x]    [2]
A [y] = [3]
    [z]    [6]

@ [x]            [2]    [ 2  6   0 ] [2]          [1]
 ˆ [y] = A -1 [3] = [ 1 -8 11 ] [3]/22 = [2]
     [z]            [6]    [ -3 2 11 ] [6]          [3]
 
i“š‚¦j x=1, y=2, z=3



ŽQlURLF
•s’èÏ•ªƒTƒCƒgFhttp://integrals.wolfram.com/index.jsp



(C)Copyright 2007-2014 Mathcot.H.I. Copyright Reserved.
    Update:2007.07.01
    Update: 2007.12.23
    Update: 2014.02.26


inserted
      by FC2 system inserted by FC2 system