If the equation `x^(4)-4x^(3)+ax^(2)+bx+1=0` has four positive roots, find the values of a and b.

Explanation

Let `x_(1),x_(2),x_(3),x_(4),`
are the roots of the equation `x_(1)x_(1)x_(1)x^(4)-4x^(3)+ax^(2)+bx+1=0 " " "....(i)"`
`:.x_(1)+x_(2)+x_(3)+x_(4)=4 " and "x_(1)x_(2)x_(3)x_(4)=1`
`:.AM=(x_(1)+x_(2)+x_(3)+x_(4))/(4)=(4)/(4)=1`
and `GM=(x_(1)x_(2)x_(3)x_(4))^((1)/(4))=1`
`i.e., AM=GM`
which is true only when `x_(1)=x_(2)=x_(3)=x_(4)=1`
Hence, given equation has all roots identical, equal to i.e., equation have form
`(x-1)^(4)=0`
`implies x^(4)-4x^(3)+6x^(2)-4x+1=0" " "....(ii)"`
On comparing Eqs. (i) and (ii), we get
`a=6,b=-4`.