Question

If all roots of equation X5 – 10x4 + ax + bx2 + CX - 32 = 0 are positive thenaf HHCXUI X5 – 10x4 + ax3 + bx2 + cx – 32 = 0 . HA A F14 è, ad(A) a = 20(B) a = 40(C) C = 80(D) b = -80​

Answer 1

Given :  all roots of equation x⁵– 10x⁴ + ax³ + bx2 + cx - 32 = 0 are positive  

To Find : Value of a :

Solution:

Let say roots are

α₁ , α₂ ,  α₃  , α₄  ,  α₅

Sum of roots  =  - (10)/1  = 10

α₁ + α₂ +   α₃ +  α₄ +  α₅ = 10

AM of Roots = 10/5 = 2

Product of Roots  =  -(-32)/1 = 32

α₁ . α₂ .   α₃ .  α₄ .  α₅ = 32

GM of roots =  $\sqrt[5]{32}$  = 2

AM = GM , only possible when roots are Equal

Hence α₁ = α₂ =  α₃  = α₄  =  α₅  = 2

α₁α₂ + α₁α₃ + α₁α₄ + α₁α₅ +  α₂α₃ + α₂α₄ + α₂α₅ + α₃α₄ + α₃α₅ + α₄₅  =  a/1

=> (2)(2) +(2)(2) +(2)(2) +(2)(2) +(2)(2) +(2)(2) +(2)(2) +(2)(2) +(2)(2) +(2)(2) = 1

=> 4 + 4 +4 + 4 +4 + 4 +4 + 4 +4 + 4  = 1

=> 40 = a

Value of a = 40

All roots are 2  hence

(x  - 2)⁵ = x⁵ - 10x⁴ + 40x³ - 80x² + 80x - 32

a = 40    b = - 80   c =  80

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