Let `alpha_(1),alpha_(2),......alpha_(n)` are roots of `a_(0)x^(n)+a_(1)x^(n-1)+a_(2)x^(n-2)+....+a_(n)=0` then `s_(3)=?` (`s_n` = sum of roots taken n at a time)

Differentiate |x|+a_(0)x^(n)+a_(1)x^(n-1)+a_(2)x^(n-1)+...+a_(n-1)x+a_(n)

(1+x)^(n)=a_(0)+a_(1)x+a_(2)*x^(2)+......+a_(n)x^(n) then prove that

a_(0)f^(n)(x)+a_(1)f^(n-1)(x)*g(x)+a_(2)f^(n-2)g^(2)(x)+......+a_(n)g^(n)(x)>=0

Differentiate the following with respect of x:a_(0)x^(n)+a_(1)x^(n-1)+a_(2)x^(n-2)++a_(n-1)x+a_(n)

Let f(x)=a_(0)x^(n)+a_(1)x^(n-1)+a_(2)x^(n-2)+......+a_(n),(a_(0)!=0) if a_(0)+a_(1)+a+_(2)+......+a_(n)=0 then the root of f(x) is

(1+x)^(n)=a_(0)+a_(1)x+a_(2)x^(2) +......+a_(n)x^(n) then Find the sum of the series a_(0) +a_(2)+a_(4) +……

Let nge3 be an integer. For a permutaion sigma=(a_(1),a_(2),.....,a_(n)) of (1,2,.....,n) we let f_(sigma)(x)=a_(n)X^(n-1)+a_(n-1)X^(x-2)+....+a_(2)x+a_(1) . Let S_(sigma) be the sum of the roots of f_(sigma)(x)=0 and let S denote the sum over all permutations sigma of (1,2,.....,n) of the numbers S_(sigma) . Then-

If a_(1),a_(2),a_(3)...,a_(n) are real numbers with a_(n)!=0 and cos alpha+i sin alpha is a root of z^(n)+a_(1)z^(n-1)+a_(2)z^(n-2)+...+a_(n-1)z+a_(n) then the value of a_(1)cos alpha+a_(2)cos2 alpha+a_(3)cos3 alpha+...

. If a_(1),a_(2),a_(3),...,a_(2n+1) are in AP then (a_(2n+1)+a_(1))+(a_(2n)+a_(2))+...+(a_(n+2)+a_(n)) is equal to