Roots of an equation

Question

If the equation
!!(x^4 - 4x^3 + ax^2 + bx + 1)!! !!=!! !!0!! has only real roots.Then the value of !!a!! and !!b!!,
a)!!(6,4)!!
b)!!(6,-4)!!
c)!!(4,-6)!!
d)!!(-4,6)!!

Solution:

Since it is given that the roots of the equation are real can say !!x_1!!,!!x_2!!,!!x_3!! and !!x_4!! be the roots of the equation.

So
!!(x_1 + x_2 + x_3 + x_4)!! !!=!! !!4!! and !!(x_1 xx x_2 xx x_3 xx x_4)=1!!

Now applying AM GM inequality we have

!!(x_1 + x_2 + x_3 + x_4)/4=(x_1x_2x_3x_4)!!

This case occurs whenever !!x_1=x_2=x_3=x_4=1!!
So,the equation becoems;
!!(x-1)^4=0!!

By binomial expansion,we get;
!!a=6!! and !!b=(-4)!!

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