Polynomials

(a i) (x - α)(x - β)(x- γ) = x + px + qx + r
ln (x - α) + ln (x - β) + ln (x- γ) = ln(x + px + qx + r)
Diff. both sides w.r.t. x:
1/(x - α) + 1/(x - β) + 1/(x- γ) = (3x + 2px + q)/(x + px + qx + r)
(ii) From (i), multiplying x + px + qx + r to both sides gives:
(x - β)(x- γ) + (x - α)(x- γ) + (x - α)(x - β) = 3x + 2px + q
Sub x = α:
(α - β)(α- γ) = 3α + 2pα + q
(b i) By division alg, we have:
(3x + 2px + q)f(x) = (x + px + qx + r)Q(x) + (Ax + Bx + C)
= (x - α)(x - β)(x- γ)Q(x) + (Ax + Bx + C)
where Q(x) is some polynomial with real coeff. Thus:
(3α + 2pα + q)f(α) = Aα + Bα + C
f(α) = (Aα + Bα + C)/(3α + 2pα + q)
(3β + 2pβ + q)f(β) = Aβ + Bβ + C
f(β) = (Aβ + Bβ + C)/(3β + 2pβ + q)
(3γ + 2pγ + q)f(γ) = Aγ + Bγ + C
f(γ) = (Aγ + Bγ + C)/(3γ + 2pγ + q)
Let (Ax + Bx + C)/(x + px + qx + r) = a/(x - α) + b/(x - β) + c/(x- γ)
Then:
(Ax + Bx + C)/(x + px + qx + r) = [a(x - β)(x- γ) + b(x - α)(x- γ) + c(x - α)(x - β)]/[(x - α)(x - β)(x- γ)]
Ax + Bx + C = a(x - β)(x- γ) + b(x - α)(x- γ) + c(x - α)(x - β)
By putting x = α, we have:
Aα + Bα + C = a(α - β)(α- γ)
a = (Aα + Bα + C)/[(α - β)(α- γ)] = (Aα + Bα + C)/(3α + 2pα + q) (From a ii)
= f(α)
Similarly, putting x = β and x = γ will give b = f(β) and c = f(γ) respectively.
So:
f(α)/(x - α) + f(β)/(x - β) + f(γ)/(x - γ) = (Ax + Bx + C)/(x + px + qx + r)
(ii) From (i), we have:
(Ax + Bx + C)/(x + px + qx + r) = f(α)/(x - α) + f(β)/(x - β) + f(γ)/(x - γ)
= [(x - β)(x- γ)f(α) + (x - α)(x- γ)f(β) + (x - α)(x - β)f(γ)]/[(x - α)(x - β)(x- γ)]
Ax + Bx + C = (x - β)(x- γ)f(α) + (x - α)(x- γ)f(β) + (x - α)(x - β)f(γ)
= [f(α) + f(β) + f(γ)]x - [(β+ γ)f(α) + (γ+ α)f(β) + (α+ β)f(γ)]x + [βγf(α) + γαf(β) + αβf(γ)]
Hence by comparing coeff on both sides:
A = f(α) + f(β) + f(γ)
B = (β+ γ)f(α) + (γ+ α)f(β) + (α+ β)f(γ)
C = βγf(α) + γαf(β) + αβf(γ)

2009-07-19 11:04:27 補充：
STEVIE-G™ 你都好好野, 邊年題目都記得!!!

2009-07-19 11:07:43 補充：
呢個 platform 真係唔多店, 唔單止個次方 show 唔到, 個負號都無啦啦同我剷埋:
B = - [(β+ γ)f(α) + (γ+ α)f(β) + (α+ β)f(γ)]
希望今次 show 到

佢考個年 xd check check 先xd

B 要加上個負號?

heehee,六呎將軍很熟吧,你考那年的題目!