Find the equation whose roots are the squares of the roots of x* + ax + bx² + cx + d = 0 and if a,B,y and s are the roots, find the values of i) Σα' i) Σαβ iii) Σαβγό
Answers
Answer:
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Step-by-step explanation:
Let α,β,γ be the roots of x3−ax2+bx−1=0, then
S1=α+β+γ=a
S2=αβ+αγ+βγ=b
S3=αβγ=1
Then α2+β2+γ2=(α+β+γ)2−2(αβ+αγ+βγ)=a2−2b
α2β2+α2γ2+β2γ2=(αβ+αγ+βγ)2−2αβγ(α+β+γ)=b2−2a
α2β2γ2=(αβγ)2=1
Hence, equation whose roots is α2,β2,γ2 is
x3−x
Answer:
Let α,β,γ be the roots of x
3
−ax
2
+bx−1=0, then
S
1
=α+β+γ=a
S
2
=αβ+αγ+βγ=b
S
3
=αβγ=1
Then α
2
+β
2
+γ
2
=(α+β+γ)
2
−2(αβ+αγ+βγ)=a
2
−2b
α
2
β
2
+α
2
γ
2
+β
2
γ
2
=(αβ+αγ+βγ)
2
−2αβγ(α+β+γ)=b
2
−2a
α
2
β
2
γ
2
=(αβγ)
2
=1
Hence, equation whose roots is α
2
,β
2
,γ
2
is
x
3
−x
2
(a
2
−2b)+x(b
2
−2a)−1=0
As both the equations is identical then
(a
2
−2b)=ua and (b
2
−2a)=vb
Hence, a=b=0; a=b=3 and a+b=−1
From a=b=3 and a+b=−1
ab=2
Therefore, option (B),(C) and (D) are true.