Demonstrate if ( 1 - x1 )( 1 - x2 )( 1 - x3 )( 1 - x4 ) = 5 where x1,2,3,4 roots of the equation .

Let us supoose that  the required 4th degree equation is x^4+ax^3+bx^2+cx+d, whose roots are x1,x2,x3 and x4,

Then sum of roots  =-a

sum of product of roots taken 2 at a time = b

sum of the product of roots taken 3 at a time = -c

prduct of roots = d

Now (1-x1)((1-x2)(1-x3)(1-x4) = d. Expanding the LHS, we get:

1-(sum of the roots)+(product of roots taken 2 at a time)- product of roots taken 3 at a time) + oroduct of all 4 roots = 5. Or

1-(-a)+b-(-c)+d = 5. Or

1+a+b+c+d = 5. Or

a+b+c+d = 4 is the condition.

Example for the demonstration:

(x-2)^3 *(x-6) = o has roots x1=x2=x3= 2 and x4 = 6

Then (x-2)^3(x-6) = x^4 -12x^3+48x^2-80x+48 = 0.

a= -12,b =48,c =-80 and d = 48.

(a+b+c+d) = -12+48-80+48 = 4.

(1-x1)(1-x2)(1-x3)(1-x4) = (1-2)(1-2)(1-2)(1-6) = (-1)(-1)(-1)(-5) = 5.

So the there exists the 4th degree equation, x^4+ax^3+bx^3+cx+d = 0,whosex1,x2,x3 and x4 can satisfy (1-x1)(1-x2)(1-x3)(1-x4) = 5, if a+b+c+d = 4.