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If the eqn x 4 - 4x 3 +a x 2 +bx +1 = 0 has four positive roots, then show that a=6 and b=-4.

If the eqn x4 - 4x3 +a x2 +bx +1 = 0 has four positive roots, then show that a=6 and b=-4.

Grade:12

1 Answers

Ramesh V
70 Points
13 years ago

since all roots are positive,

coeff. of x shld be negatuve i.e., b is negative

let roots be m(>0), n(>0), p(>0), q(>0)

m+n+p+q = 4

mn+np+pq+mp+mq+nq = a

mnp+mnq+npq+mpq = -b

mnpq = 1

For the objective case,

put m=n=p=q =1 (which satisfies eqns 1 and 4 above)

so a = 1+1+1+1+1+1 = 6

    -b = 1+1+1+1 =4

so, a = 6 and b = -4

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