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

Options

A. a=6, b=-4
B. a=-4. b=6
C. a=-6, b=4
D. a=-6, b=-4

Answer

A

Explanation

Let `x_(1), x_(2), x_(3)` and `x_(4)`, be four postive roots of the given equation. Then,
`x_(1)+x_(2)+x_(3)+x_(4)=4" and "x_(1)x_(2)x_(3)x_(4)=1`
`implies" "(x_(1)+x_(2)+x_(3)+x_(4))/(4)=1" and "(x_(1)x_(2)x_(3)x_(4))^(1//4)=1`
`impliesA.M." of "x_(1),x_(2),x_(3),x_(4)" is equal to their "G.M.`
`implies" "x_(1)=x_(2)=x_(3)=x_(4)`
`implies" "x_(1)=x_(2)=x_(3)=x_(4)=1" "[becausex_(1)+x_(2)+x_(3)+x_(4)=4]`
`:." "x^(4)-4x^(3)+ax^(2)+bx+1=(x-1)^(4)`
`implies" "x^(4)-4x^(3)+ax^(2)+bx+a=x^(4)-4x^(3)+6x^(2)-4x+1`
`implies" "a=6" and "b=-4`