bçfÈ 31 » 1 ‚, pp. 46-48
j˙ 5;í n Ÿj5¸íj¶
"‰¤
ÊÅ2bç£ò2bç`‡2, %}|Û¥5tæñ:
ùŸj˙ x2−4x + 5 = 0 5s;Ñ α D β
t° (1) α + β (2) α2+ β2 (3) α3+ β3 5M
˛Fíj¶à-:
Ä ax2+ bx + c = 0 (a 6= 0) ís;¸Ñ −b
a, s; Ñ c a, † (1) α + β = 4 / α · β = 5
(2) α2+ β2 = (α + β)2−2αβ = 16 − 10 = 6
(3) α3+ β3 = (α + β)3−3αβ(α + β) = 64 − 60 = 4 DP4T|-j¶:
I f (x) = x2−4x + 5 † f0(x) = 2x − 4
‚à f0(x) ÷ f (x) Vj,Wí 3 _½æ
2 4 6 4 −14 · · · · 1 −4 5 ) 2 −4
2 −8 10
4 −10 0 4 −16 20
6 −20 0 6 −24 30
4 −30 0 4 −16 20
−14 −20 0
−14 56 −70
· · · ·
46
j˙ 5;í n Ÿj5¸íj¶ 47
*¼ í 2_båÇá, ) (1) α + β = 4 (2) α2+ β2 = 6 (3) α3+ β3 = 4 _!! ß ÿJ! 7/¤j¶´ªJl| α4+ β4 = −14, · · · 0Vð¤ã¿u´£ü?
α4+ β4 = (α2+ β2)2−2α2β2 = 36 − 50 = −14 á….Ï!
¥Ôyj¶SÜ;W? ?´RBúŸj˙ CyòŸj˙ ý? çÍuí
ùŸj˙ í;Ñ α D β, .ÜøO4, q¤j˙ Ñ x2−(α + β)x + αβ = 0
I f (x) = x2−(α + β)x + αβ = (x − α)(x − β) / f0(x) = 2x − (α + β)
† f0(x)
f(x) = 2x − (α + β)
(x − α)(x − β) = 1
x− α + 1 x− β
Ä 1 x− α =
1 x 1 − α
x
= 1 x ×h
1 +α x
+α x
2 +α
x
3
+ · · · +α x
n
+ · · ·i
= 1 x + α
x2 +α2 x3 + α3
x4 + · · · + αn
xn+1 + · · · / 1
x− β = 1 x 1 − β
x
= 1 x ×h
1 +β x
+β x
2 +β
x
3
+ · · · +β x
n
+ · · ·i
= 1 x + β
x2 +β2 x3 +β3
x4 + · · · + βn
xn+1 + · · · FJ f0(x)
f(x) =
∞
X
k=0
αk+ βk xk+1 Ĥ, *øÇáíWæíÔyj¶
2x − 4
x2−4x + 5 = 2 x + 4
x2 + 6 x3 + 4
x4 +−14 x5 + · · ·
)ƒ (1) α + β = 4 (2) α2+ β2 = 6 (3) α3+ β3 = 4 (4) α4+ β4 = −14, . . . .
k×)ê1í!
yVnúŸj˙ í8$
JúŸj˙ í;Ñ α, β D γ, .ÜøO4, q¤j˙ Ñ (x − α)(x − β)(x − γ) = 0
I f (x) = x3−(α + β + γ)x2+ (αβ + αγ + βγ)x − αβγ = (x − α)(x − β)(x − γ)
48 bçfÈ 31 » 1 ‚ ¬ 96 3 ~
/ f0(x) = 3x2−2(α + β + γ)x + (αβ + αγ + βγ)
†f0(x)
f(x) = 3x2 −2(α + β + γ)x + (αβ + αγ + βγ) (x − α)(x − β)(x − γ)
= 1
x− α + 1
x− β + 1 x− γ
=
∞
X
k=0
αk+ βk+ γk xk+1
°Ü, J n Ÿj˙ í;Ñ αi, i = 1, 2, . . . , n, .ÜøO4, q¤j˙ Ñ
n
Y
i=1
(x − αi) = 0
I f (x) =
n
Y
i=1
(x − αi)
/ f0(x) =
n
X
i=1 n
Y
j6=i
(x − αi) =
n
X
i=1
(x − α1)(x − α2) · · · (x − αn) x− αi
† f0(x) f(x) =
n
X
i=1
1 x− αi =
∞
X
k=1 n
X
i=1
αki xk+1 Ĥ, %(° αn+ βn+ γn v, Uà f0(x)
f(x) Vjæ, ÿ.Ûb‚àyÅí°Mt 7!!
