Question

Find the minimum value of a+b+c+d in the equation $x^5-a{x}^4+b{x}^3-c{x}^2+dx-243=0,$ if it is given that the roots are positive real numbers ?

• A. 780
• B. 660
• C. 30
• D. 720

Answer 1

The correct option is A

780

There are 5 roots in this equation.

Here the product of the roots is given as 243. Hence the sum will be minimum when the roots are equal. In fact a,b,c and d will be minimum when the roots are equal.

243 = $3^5$

Hence, the roots will be 3,3,3,3 and 3

a= sum of the roots = 15

b=Sum of product of roots taken 2 at a time = (3 $\times$ 3)+(3 $\times$ 3)...10 times or = ${}^5{C}_2\times 9$ = 90

c= Sum of product of roots taken 3 at a time = (3 $\times$ 3$\times$ 3)+(3$\times$ 3$\times$ 3)...5 times or = ${}^5{C}_3\times 27$ = 270

d=Sum of product of roots taken 4 at a time = (3$\times$ 3$\times$ 3$\times$ 3)....5 times= 405

Hence, a+b+c+d = 780