In the prime factorization of 37!=`2^(alpha_(2)).3^(alpha_(3)).5^(alpha_(5))......37^(alpha_(37))` then the ratio `alpha_(3):alpha_(5)=`

In the prime factorization of 37! = 2^(a_(1)).3^(a_(2)).5^(a_(3))....37^(a_(n)) the ratio alpha_(3):alpha_(5) =

If the roots of equation x^(3) + ax^(2) + b = 0 are alpha _(1), alpha_(2), and alpha_(3) (a , b ne 0) . Then find the equation whose roots are (alpha_(1)alpha_(2)+alpha_(2)alpha_(3))/(alpha_(1)alpha_(2)alpha_(3)), (alpha_(2)alpha_(3)+alpha_(3)alpha_(1))/(alpha_(1)alpha_(2)alpha_(3)), (alpha_(1)alpha_(3)+alpha_(1)alpha_(2))/(alpha_(1)alpha_(2)alpha_(3)) .

Let |(1+x,x,x^(2)),(x,1+x,x^(2)),(x^(2),x,1+x)|=1/6(x-alpha_(1))(x-alpha_(2))(x-alpha_(3))(x-alpha_(4)) be an indentity in x , where alpha_(1),alpha_(2),alpha_(3),alpha_(4) are independent of x . Then find the value of alpha_(1)alpha_(2)alpha_(3)alpha_(4) .

let |{:(1+x,x,x^(2)),(x,1+x,x^(2)),(x^(2),x,1+x):}|=(1)/(6)(x-alpha_(1))(x-alpha_(2))(x-alpha_(3))(x-alpha_(4)) be an identity in x, where alpha_(1),alpha_(2),alpha_(3),alpha_(4) are independent of x. Then find the value of alpha_(1)alpha_(2)alpha_(3)alpha_(4)

If 1,alpha_(1),alpha_(2),alpha_(3),alpha_(4) are the fifth roots of unity then sum_(i=1)^(4)(1)/(2-alpha_(i))=

If |{:(1+x^(2),3x,x+1),(5,x,x^(3)),(0,1,x+2):}|=(alpha_(1)-x)(alpha_(2)-x)(alpha_(3)-x)(alpha_(4)-x)(alpha_(5)-x) is an identity, where alpha_(1),alpha_(2),alpha_(3),alpha_(4),alpha_(5) are complex numbers independent of x ,then the value of alpha_(1)*alpha_(2)*alpha_(3)*alpha_(4)*alpha_(5) is

If alpha_(2)

If alpha_(1),alpha_(2),alpha_(3)......alpha_(n) are roots of the equation f(x)=0 then (-alpha_(1),-alpha_(2),-alpha_(3),......alpha_(n)) are the roots of