Question

if alpha, beta, gamma are the roots of equation x³ - 4x² + x + 6 then the equation roots are 1/alpha, 1/beta , 1/gamma is



​

Answer 1

Answer:

Step-by-step explanation:

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