Question

hong

想請問這題如何解

8. 設f(x)=x+ax²+bx²+cx+d為一實係數多項式,已知f(x)=0沒有實根,設a,B,y,8為f(x)=0的四個複數根,
若a+B=-1+4i且y.8=-1+17i,其中=I,則序組(a,b,c,d) = (2,15,134290)
(athi) + (c+li)
5+31
4727
5-3分
4-2λ

複數

Answer 1

✨ Jawaban Terbaik ✨

qn

α、β非共軛複數
α跟γ或δ為共軛複數

設 α跟γ、β跟δ為共軛複數
α = p+qi
γ = p-qi
β = r+si
δ = r-si

α+β = (p+r)+(q+s)i = -1+4i
⇒ γ+δ = (p+r)-(q+s)i = -1-4i
γδ = (pr-qs)-(ps+qr)i = -1+17i
⇒ αβ = (pr-qs)+(ps+qr)i = -1-17i

a = -(α+β+γ+δ) = -(-1+4i + -1-4i) = 2

b = αβ+αγ+αδ+βγ+βδ+γδ
= αβ + γδ + (α+β)(γ+δ)
= (-1-17i) + (-1+17i) + (-1+4i)(-1-4i)
= -2 + 17
= 15

c = - ( αβγ + αβδ + αγδ + βγδ )
= - [ αβ(γ+δ) + (α+β)γδ ]
= - [(-1-17i)(-1-4i) + (-1+4i)(-1+17i)]
= - [(-67+21i) + (-67-21i)]
= 134

d = αβγδ
= (-1-17i)(-1+17i)
= 290

hong

謝謝！