Question

if alpha,beta,gamma,delta are the roots of x^4- 4x^3 + 5x 2- 2 x-2= 0,then product of the roots is

Answer 1

p(x)=x

4

+4x

3

+5x

2

+2x−2=0

If one of the roots id (−1+i) then (−1−i) is alos a root of p(x)

⇒{x−(−1+i)}{x−(−1−i)} is a factor of p(x)

⇒(x+1)

2

−(i)

2

is a factor of p(x)

⇒x

2

+2x+2 is a factor of p(x)

Dividing p(x) by x

2

+2x+2

⇒p(x)=(x

2

+2x−1)(x

2

+2x+2)

⇒(x

2

+2x−1)(x

2

+2x+2)

⇒x

2

+2x−1=0

⇒x=

2

−2±

4−4(1)(−1)

⇒x=

2

−2±2

2

⇒x=−1±

2

So the other two roots are −1+

2

and −1−

2