If a, ß, y are zeroes of
33 – 2α2 + 50 – 6, then
αβ + βγ + γα
Answers
If α,β,γ are the roots of x
3
−6x−4=0, then the equation whose roots are βγ+
α
1
,γα+
β
1
,αβ+
γ
1
is
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ANSWER
Given equation: x
3
−6x−4=0 roots of the given equation is given by
α+β+γ=0
αβ+βγ+γα=−6
αβγ=4
now the value of roots given in the question are
αβ+
γ
1
=
γ
αβγ+1
=
γ
5
αγ+
β
1
=
β
αβγ+1
=
β
5
βγ+
α
1
=
α
αβγ+1
=
α
5
Now calculating the sum of the roots, we get
γ
5
+
β
5
+
α
5
=
2
−15
Now calculating product of the roots, we get
γ
5
∗
β
5
∗
α
5
=
4
125
Now calculating sum of the products of the roots, we get
γ
5
∗
β
5
+
β
5
∗
α
5
+
γ
5
∗
α
5
=0
Therefore the general cubic equation is
x
3
+(Sum of the roots)x
2
+(sum of the products of the roots)x+(products of the roots)=0
x
3
+(
2
−15
)x
2
+(0)x+(
4
125
)=0
4x
3
−30x
2
+125=0 which is the required equation.